a measurement of the cosmic microwave background … · the spt measurement of the cmb damping tail...

The Astrophysical Journal, 779:86 (19pp), 2013 December 10 doi:10.1088/0004-637X/779/1/86C© 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

A MEASUREMENT OF THE COSMIC MICROWAVE BACKGROUND DAMPING TAIL FROM THE2500-SQUARE-DEGREE SPT-SZ SURVEY

K. T. Story1,2, C. L. Reichardt3, Z. Hou4, R. Keisler1,2, K. A. Aird5, B. A. Benson1,6, L. E. Bleem1,2,J. E. Carlstrom1,2,6,7,8, C. L. Chang1,6,8, H.-M. Cho9, T. M. Crawford1,7, A. T. Crites1,7, T. de Haan10, M. A. Dobbs10,

J. Dudley10, B. Follin4, E. M. George3, N. W. Halverson11, G. P. Holder10, W. L. Holzapfel3, S. Hoover1,2,J. D. Hrubes5, M. Joy12, L. Knox4, A. T. Lee3,13, E. M. Leitch1,7, M. Lueker14, D. Luong-Van5, J. J. McMahon15, J. Mehl1,8,

S. S. Meyer1,2,6,7, M. Millea4, J. J. Mohr16,17,18, T. E. Montroy19, S. Padin1,7,14, T. Plagge1,7, C. Pryke20, J. E. Ruhl19,J. T. Sayre19, K. K. Schaffer1,6,21, L. Shaw10, E. Shirokoff3, H. G. Spieler13, Z. Staniszewski19, A. A. Stark22,

A. van Engelen10, K. Vanderlinde23,24, J. D. Vieira14, R. Williamson1,7, and O. Zahn251 Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA; [email protected]

2 Department of Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA3 Department of Physics, University of California, Berkeley, CA 94720, USA

4 Department of Physics, University of California, One Shields Avenue, Davis, CA 95616, USA5 University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

6 Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA7 Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

8 Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA9 NIST Quantum Devices Group, 325 Broadway, Mailcode 817.03, Boulder, CO 80305, USA

10 Department of Physics, McGill University, 3600 Rue University, Montreal, Quebec H3A 2T8, Canada11 Department of Astrophysical and Planetary Sciences and Department of Physics, University of Colorado, Boulder, CO 80309, USA

12 Department of Space Science, VP62, NASA Marshall Space Flight Center, Huntsville, AL 35812, USA13 Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

14 California Institute of Technology, MS 249-17, 1216 E. California Boulevard, Pasadena, CA 91125, USA15 Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, MI 48109, USA

16 Department of Physics, Ludwig-Maximilians-Universitat, Scheinerstr. 1, D-81679 Munchen, Germany17 Excellence Cluster Universe, Boltzmannstr. 2, D-85748 Garching, Germany

18 Max-Planck-Institut fur extraterrestrische Physik, Giessenbachstr., D-85748 Garching, Germany19 Physics Department, Center for Education and Research in Cosmology and Astrophysics, Case Western Reserve University, Cleveland, OH 44106, USA

20 Department of Physics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA21 Liberal Arts Department, School of the Art Institute of Chicago, 112 S. Michigan Avenue, Chicago, IL 60603, USA

22 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA23 Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada

24 Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada25 Berkeley Center for Cosmological Physics, Department of Physics, University of California and

Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAReceived 2012 October 25; accepted 2013 October 9; published 2013 November 26

ABSTRACT

We present a measurement of the cosmic microwave background (CMB) temperature power spectrum using datafrom the recently completed South Pole Telescope Sunyaev–Zel’dovich (SPT-SZ) survey. This measurement ismade from observations of 2540 deg2 of sky with arcminute resolution at 150 GHz, and improves upon previousmeasurements using the SPT by tripling the sky area. We report CMB temperature anisotropy power over themultipole range 650 < � < 3000. We fit the SPT bandpowers, combined with the 7 yr Wilkinson MicrowaveAnisotropy Probe (WMAP7) data, with a six-parameter ΛCDM cosmological model and find that the two datasetsare consistent and well fit by the model. Adding SPT measurements significantly improves ΛCDM parameterconstraints; in particular, the constraint on θs tightens by a factor of 2.7. The impact of gravitational lensing isdetected at 8.1σ , the most significant detection to date. This sensitivity of the SPT+WMAP7 data to lensing by large-scale structure at low redshifts allows us to constrain the mean curvature of the observable universe with CMB dataalone to be Ωk = −0.003+0.014

−0.018. Using the SPT+WMAP7 data, we measure the spectral index of scalar fluctuationsto be ns = 0.9623 ± 0.0097 in the ΛCDM model, a 3.9σ preference for a scale-dependent spectrum with ns < 1.The SPT measurement of the CMB damping tail helps break the degeneracy that exists between the tensor-to-scalarratio r and ns in large-scale CMB measurements, leading to an upper limit of r < 0.18 (95% C.L.) in the ΛCDM+rmodel. Adding low-redshift measurements of the Hubble constant (H0) and the baryon acoustic oscillation (BAO)feature to the SPT+WMAP7 data leads to further improvements. The combination of SPT+WMAP7+H0+BAOconstrains ns = 0.9538 ± 0.0081 in the ΛCDM model, a 5.7σ detection of ns < 1, and places an upper limit ofr < 0.11 (95% C.L.) in the ΛCDM+r model. These new constraints on ns and r have significant implications forour understanding of inflation, which we discuss in the context of selected single-field inflation models.

Key words: cosmic background radiation – cosmology: observations – large-scale structure of universe

Online-only material: color figures

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http://dx.doi.org/10.1088/0004-637X/779/1/86

mailto:[email protected]

The Astrophysical Journal, 779:86 (19pp), 2013 December 10 Story et al.

1. INTRODUCTION

Over the past two decades, measurements of the cosmicmicrowave background (CMB) have provided profound insightinto the nature of the universe. Detailed information about thecomposition and evolution of the universe is encoded in thetemperature and polarization anisotropy of the CMB. Measuringthis anisotropy enables powerful tests of cosmological theory.On degree scales, CMB anisotropy is generated primarilyby the acoustic oscillations of the primordial plasma in theearly universe. The Wilkinson Microwave Anisotropy Probe(WMAP) satellite has been used to measure these acousticoscillations with cosmic variance-limited precision on angularscales corresponding to � � 500 (Komatsu et al. 2011, hereafterWMAP7). On much smaller angular scales, primary CMBanisotropy becomes dominated by effects imprinted on theCMB at low redshift (so-called secondary anisotropy) andforegrounds; at millimeter wavelengths, this transition occursat � ∼ 3000. This small, angular-scale millimeter-wavelengthanisotropy has been measured by the South Pole Telescope(SPT; Lueker et al. 2010; Shirokoff et al. 2011; Reichardt et al.2012) and the Atacama Cosmology Telescope (ACT; Fowleret al. 2010; Das et al. 2011a).

The anisotropy in the CMB at intermediate angular scales,1000 � � � 3000, is often referred to as the “damping tail”because the anisotropy power on these angular scales is dampedby photon diffusion during recombination (Silk 1968). Addingmeasurements of the damping tail to large-scale CMB mea-surements considerably tightens the resulting cosmological con-straints. The wider range of angular scales also enables betterconstraints on the sound horizon at recombination (by measur-ing more acoustic peaks) and the slope of the primordial powerspectrum. Last, although tensor perturbations from cosmic infla-tion add CMB power only at very large angular scales, the effectof these tensor perturbations is degenerate with changes in ns inlarge-scale measurements. Damping-tail measurements helpbreak this degeneracy, thus tightening constraints on the levelof tensor perturbations.

In the past few years, there have been several increasinglyprecise measurements of the CMB damping tail, including theArcminute Cosmology Bolometer Array Receiver (ACBAR;Reichardt et al. 2009), QUaD (Brown et al. 2009; Friedmanet al. 2009), ACT (Das et al. 2011b), and SPT (Keisler et al.2011, hereafter, K11). The most precise published measurementof the CMB damping tail before this work comes from thefirst 790 deg2 of the South Pole Telescope Sunyaev–Zel’dovich(SPT-SZ) survey (K11).

In this paper, we present a measurement of the power spec-trum from the third acoustic peak through the CMB dampingtail, covering the range of angular scales corresponding to mul-tipoles 650 < � < 3000. This power spectrum is calculatedfrom the complete SPT-SZ survey covering 2540 deg2 of sky,and improves upon the results presented in K11 by expandingthe sky coverage by a factor of three.

We present constraints from this measurement on the standardΛCDM model of cosmology, then extend the model to quantifythe amplitude of gravitational lensing of the CMB. We use thissensitivity to gravitational lensing by large-scale structure atlow redshifts to measure the mean curvature of the observableuniverse from CMB data alone. We also consider modelsincluding tensor perturbations and explore implications of theresulting parameter constraints for simple models of inflation.Adding low-redshift measurements of the Hubble constant (H0)

2h 0h

22h4h

6h

-40°-50°

-60°

Figure 1. 2500 deg2 SPT-SZ survey. We show the full survey region withlightly filtered 95 GHz data from the SPT, using the data and filters that bestcapture the degree-scale anisotropy of the CMB visible in this figure. The powerspectrum measurement reported in this paper is calculated from 2540 deg2 ofsky and analyzes 150 GHz data with a different high-pass filter, as described inSection 2.2.

(A color version of this figure is available in the online journal.)

and the baryon acoustic oscillation (BAO) feature to the CMBdata further tightens parameter constraints, and we presentcombined parameter constraints for each of the aforementionedmodel extensions. Implications of the SPT power spectrum fora larger range of extensions to the standard cosmological modelare explored more fully in a companion paper, Hou et al. (2012,hereafter H12).

This paper is organized as follows. We describe the SPTobservations and data reduction in Section 2. We present thepower spectrum calculation in Section 3. We discuss testsfor systematic errors in Section 4.1. We present the powerspectrum measurement in Section 5. In Section 6, we outlineour cosmological parameter fitting framework and present theresulting parameter constraints, then use these constraints toexplore the implications for simple models of inflation. Weconclude in Section 7.

2. OBSERVATIONS AND DATA REDUCTION

The SPT is a telescope with a 10 m diameter and is locatedat the Amundsen-Scott South Pole station in Antarctica. Thefirst survey with the SPT, referred to as the “SPT-SZ” survey,was completed in 2011 November and covered a ∼2500 deg2

region of sky between declinations of −40◦ and −65◦ and rightascensions (R.A.s) of 20 hr and 7 hr. The SPT-SZ survey isshown in Figure 1. Here, we present the first power spectrummeasurement that uses data from the complete SPT-SZ survey.We use data from 2540 deg2 of sky in this analysis.

This work uses observations and data reduction methods thatare very similar to those described in K11. In this section,we give an overview of the observations and data reduction,highlighting the differences with the treatment in K11, to whichwe refer the reader for a detailed treatment of the analysismethods.

2.1. Observing Strategy and Fields

From 2008 to 2011, the SPT was used to observe a contiguous∼2500 deg2 patch of sky to a noise level of approximately

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Table 1The Fields Observed with the SPT between 2008 and 2011

Name R.A. Decl. ΔR.A. ΔDecl. Effective Area(◦) (◦) (◦) (◦) (deg2)

ra5h30dec-55 82.7 −55.0 15 10 84ra23h30dec-55 352.5 −55.0 15 10 83ra21hdec-60 315.0 −60.0 30 10 155ra3h30dec-60 52.5 −60.0 45 10 227ra21hdec-50 315.0 −50.0 30 10 192ra4h10dec-50 62.5 −50.0 25 10 156ra0h50dec-50 12.5 −50.0 25 10 157ra2h30dec-50 37.5 −50.0 25 10 157ra1hdec-60 15.0 −60.0 30 10 152ra5h30dec-45 82.5 −45.0 15 10 109ra6h30dec-55 97.5 −55.0 15 10 85ra23hdec-62.5 345.0 −62.5 30 5 75ra21hdec-42.5 315.0 −42.5 30 5 121ra22h30dec-55 337.5 −55.0 15 10 84ra23hdec-45 345.0 −45.0 30 10 217ra6hdec-62.5 90.0 −62.5 30 5 75ra3h30dec-42.5 52.5 −42.5 45 5 179ra1hdec-42.5 15.0 −42.5 30 5 119ra6h30dec-45 97.5 −45.0 15 10 111

Total 2540

Notes. The locations and sizes of the fields observed by the SPT between 2008 and 2011. For eachfield, we give the center of the field in right ascension (R.A.) and declination (decl.), the nominalextent of the field in R.A. and decl., and the effective field area as defined by the window (seeSection 3.2).

18′ μK26 at 150 GHz.27 This area of sky was observed in19 contiguous sub-regions, which we refer to as observation“fields.” In the basic survey strategy, the SPT was used to observea single field until the desired noise level was reached beforemoving on to the next field. Two fields were observed in 2008,three in 2009, five in 2010, and nine in 2011. All nine fieldsfrom 2011 were observed to partial depth in 2010 in order tosearch for massive galaxy clusters, then re-observed in 2011 toachieve nominal noise levels. The results of that bright clustersearch were published in Williamson et al. (2011). In terms ofsky area, this equates to observing 167 deg2 in 2008, 574 deg2

in 2009, 732 deg2 in 2010, and 1067 deg2 in 2011. The fields areshown in Figure 2, and the field locations and sizes are presentedin Table 1.

Both fields from 2008 (ra5h30dec-55 and ra23h30dec-55)were re-observed in later years to achieve lower than normalnoise levels. In this analysis, we use data from only one year foreach field because the beam and noise properties vary slightlybetween years. This choice simplifies the analysis withoutaffecting the results as the bandpower uncertainties remainsample variance dominated (see Section 3.5).

The SPT is used to observe each field in the following manner.The telescope starts in one corner of the observation field, slewsback and forth across the azimuth range of the field, and thenexecutes a step in elevation, repeating this pattern until the entirefield has been covered. This constitutes a single observation of

26 Throughout this work, the unit K refers to equivalent fluctuations in theCMB temperature, i.e., the temperature fluctuation of a 2.73 K blackbody thatwould be required to produce the same power fluctuation. The conversionfactor is given by the derivative of the blackbody spectrum (dBν/dT ),evaluated at 2.73 K.27 The SPT-SZ survey also includes data at 95 and 220 GHz. However, thiswork uses only 150 GHz data because this observing band is the most sensitivefor the SPT and the data from one observing band are sufficient to make highsignal-to-noise maps of the CMB anisotropy.

Figure 2. SPT was used to observe 2500 deg2 over 19 individual fields, whichare overlaid here on an orthographic projection of the IRAS 100 μm dust mapfrom Schlegel et al. (1998). These observation fields were chosen to lie inregions of low dust emission (dark red).


the field and takes from 30 minutes to a few hours, dependingon the specific field being observed. Azimuthal scan speedsvary between fields, ranging from 0.◦25 to 0.◦42 s−1 on the sky.The starting elevation positions of the telescope are dithered by

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between 0.′3 and 1.′08 to ensure uniform coverage of the regionin the final coadded map.

In four of the 2008 and 2009 fields, ra23h30dec-55, ra21hdec-60, ra3h30dec-60, and ra21hdec-50, observa-tions were conducted with a “lead-trail” strategy. In this ob-servation strategy, the field is divided into two halves in rightascension. The lead half is observed first, immediately followedby the trail half in a manner such that both halves are observedover the same azimuthal range. If necessary, the lead–trail datacould be analyzed in a way that cancels ground pickup. In thisanalysis, we combine lead–trail pairs into single maps, and ver-ify that contamination from ground pickup is negligible—seethe following paragraph and Section 4.1 for details.

We apply several (often redundant) data quality cuts onindividual observations using the following criteria: map noise,noise-based bolometer weight, the product of median bolometerweight with map noise, and the sum of bolometer weights overthe full map. For these cuts, we remove outliers both aboveand below the median value for each field. We do not useobservations that are flagged by one or more of these cuts. Wealso flag observations with only partial field coverage. Last,we cut maps that were made from observations in azimuthranges that could be more susceptible to ground pickup overthe angular scales of interest. We use “ground-centered” mapsto measure ground pickup on large (� ∼ 50) scales, and cutobservations that were made at the azimuths with the worst 5%ground pickup to minimize potential ground pickup on smaller,angular scales. Although this cut has an impact on our null tests(see Section 4.1), we emphasize that it does not significantlychange the power spectrum, the precision of which is limited bysample variance.

2.2. Map-making: Time-ordered Data to Maps

As the SPT scans across the sky, the response of eachdetector is recorded as time-ordered data (TOD). These TODare recorded at 100 Hz and have a Nyquist frequency of 50 Hz,which corresponds to a multipole number parallel to the scandirection (�x) between 72,000 and 43,000 at the SPT scanspeeds. Because we report the power spectrum up to only� = 3000, we can benefit computationally by reducing thesampling rate. We choose a low-pass filter and down-samplingfactor on the basis of each field’s scan speed such that they affectapproximately the same angular scales. We use a down-samplingfactor of six for 2008 and 2009, and four for 2010 and 2011,with associated low-pass filter frequencies of 7.5 and 11.4 Hz,respectively. These filtering choices remove a negligible amountof power in the signal band.

Next, the down-sampled TOD are bandpass-filtered between�x = 270 and 6600. The low-pass filter is necessary to avoidaliasing high-frequency noise to lower frequencies during map-making. The high-pass filter reduces low-frequency noise fromthe atmosphere and instrumental readout. The high-pass filter isimplemented by fitting each bolometer’s TOD (from a singleazimuthal scan across the field) to a model consisting oflow-frequency sines and cosines and a fifth-order polynomial.The best-fit model is then subtracted from the TOD. During thefiltering, we mask regions of sky within 5′ of point sources withfluxes of S150 GHz > 50 mJy. These regions are also masked inthe power spectrum analysis; see Section 3.2.

At this stage, the TOD retain signal from the atmosphere thatis correlated between detectors. We remove the correlated signalby subtracting the mean signal across each detector module for

every time sample.28 This process acts as an approximatelyisotropic high-pass filter.

The filtered TOD are made into maps using the process de-scribed by K11. The data from each detector receive a weighton the basis of the power-spectral density of that detector’scalibrated TOD in the 1–3 Hz band. This band correspondsapproximately to the signal band of this analysis. We have cal-culated the level of bias introduced by using the full (signal+ noise) power to calculate the detector weights, as opposedto using the noise power only (Dunner et al. 2013), and wefind that the level of bias is completely negligible (�0.01% inpower). The detector data are binned into maps with 1′ pix-els on the basis of the telescope-pointing information. In thepower-spectrum analysis presented in Section 3, we adopt theflat-sky approximation, whereby the wavenumber k is equiv-alent to multipole moment � and spherical harmonic trans-forms are replaced by Fourier transforms. We project from thecurved celestial sky to flat-sky maps with the oblique Lambertequal-area azimuthal projection (Snyder 1987).

2.3. Beam Functions

A precise measurement of the SPT beam—the optical re-sponse as a function of angle—is needed to calibrate the angularpower spectrum as a function of multipole. We summarize themethod used to measure the SPT beams and refer the reader toK11 or Schaffer et al. (2011) for a more detailed description.

The average 150 GHz beam is measured for each year usinga combination of maps from Jupiter, Venus, and the 18 brightestpoint sources in the CMB fields. The maps of Jupiter are usedto measure the beam outside a radius of 4′, and the maps of thebright point sources are used to measure the beam inside thatradius. Maps of Venus are used to join the inner and outer beammaps into a composite beam map. The maps of the planets arenot used to estimate the very inner beam because of a nonlineardetector response when directly viewing bright sources and thenon-negligible angular size of the planets compared to the beam.We use the composite beam map to measure the beam functionB�, defined as the azimuthally averaged Fourier transform ofthe beam map. We consider uncertainties in the measurement ofB� arising from several statistical and systematic effects, suchas residual atmospheric noise in the maps of Venus and Jupiterand account for known inter-year correlations of some of thesesources of uncertainty. The parameter constraints quoted in thiswork are not sensitive to the calculated beam uncertainties;we have tested increasing our beam uncertainties by a factorof two and have seen no significant impact on the resultingcosmological parameter fits.

A nearly identical beam treatment was used by K11. The maindifference is that the beam function was normalized to unity at� = 350 in K11 rather than � = 750 in this work. The averagemultipole of our calibration region is close to � = 750, and thischoice of normalization scale better decouples the beam andcalibration uncertainties.

2.4. Calibration

The observation-to-observation relative calibration of theTOD is determined from repeated measurements of a galacticH II region, RCW38. As in K11 and Reichardt et al. (2012),the absolute calibration is determined by comparing the SPT

28 The SPT-SZ focal plane has a hexagonal geometry with six triangularbolometer modules, each with ∼160 detectors. Each module is configured witha set of filters that determines its observing frequency of 95, 150, or 220 GHz.

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and WMAP7 power in several �-bins over the multipole range� ∈ [650, 1000]. We use the same �-bins for both experiments:seven bins with δ� = 50. This calibration method is model-independent, requiring only that the CMB power in the SPTfields is statistically representative of the all-sky power. Weestimate the uncertainty in the SPT power calibration to be 2.6%.The calibration uncertainty is included in the covariance matrix;this treatment is equivalent to including an additional calibrationparameter with a 2.6% Gaussian uncertainty in the cosmologicalparameter fits. We have also cross-checked this method againsta map-based calibration method, in which we calculate the crossspectrum between identically filtered SPT and WMAP maps over1250 deg2 of sky, and find that the calibrations between thesetwo methods are consistent, although the map-based calibrationuncertainties are larger. We do not find the parameter constraintsquoted in this work to be sensitive to the calibration uncertainty;changing the calibration uncertainty by a factor of two in eitherdirection has no significant effect on the resulting cosmologicalparameter fits.

3. POWER SPECTRUM

In this section, we describe the power-spectrum calculation.This analysis closely follows the analysis developed by Luekeret al. (2010) and used by K11; we refer the reader to those papersfor a more detailed description. We refer to the average powerover a given range of � values as a bandpower. We use a pseudo-C� method under the flat-sky approximation as described inSection 2.2. The power spectra are calculated independently ineach of the 19 fields, then combined into the final result. Wereport bandpowers in terms of D�, which is defined as follows:

D� = �(� + 1)

2πC� . (1)

To calculate D�, we use a cross-spectrum bandpower estimatoras described in Section 3.3, which has the advantage of beingfree of noise bias; see Lueker et al. (2010) for a more detaileddescription.

3.1. Maps

The basic input to the cross-spectrum estimator is a setof maps for a given field, each with independent noise. Formost fields, this input set is composed of maps from singleobservations. Each observation has statistically independentnoise because observations are temporally separated by at leastan hour and the TOD have been high-pass filtered at ∼0.2 Hz. Asin K11, for the four fields observed with a lead–trail strategy,we construct the input map set by combining lead–trail pairsinto single maps. The ra23h30dec-55 field was observed usingcomparatively large elevation steps and hence has less uniformcoverage. For this field, the single maps that are the basic inputto the cross-spectrum estimator are formed by combining twopairs of lead–trail observations. Each pair is chosen to havedifferent elevation dithers, leading to a more homogeneous fieldcoverage.

3.2. Window

For a given field, each of the maps is multiplied by the samewindow W in order to avoid sharp edges at map boundaries,control overlap between adjacent fields, and remove bright pointsources. Each window is the product of an apodization maskwith a point-source mask. The apodization masks are calculated

by applying a 1◦ taper using a Hann function to the edges ofthe uniform coverage region of each field. The observationswere designed such that the uniform coverage region overlapsbetween neighboring fields. We define our apodization windowssuch that the overlap region between adjacent fields containsa combined weight that approaches but never exceeds unity(the weight at the center of the field). This process results inapodization windows that include marginally smaller regions ofsky and fall off more slowly than the windows used in K11,which did not need to account for field overlap.

As was done in K11, we identify all point sources with150 GHz flux >50 mJy. Each of these point sources is maskedwith a 5′ radius disk that is tapered outside the disk usinga Gaussian taper with a width of σtaper = 5′. Point-sourcemasks remove 1.4% of the total sky area. Using previousmeasurements of the millimeter-wave point-source population(Vieira et al. 2010; Shirokoff et al. 2011), we estimate thatthe power from residual point sources below this flux cut isC� ∼ 1.3 × 10−5 μK2, or D� ∼ 18 μK2(�/3000)2. This poweris approximately half the CMB anisotropy power at � = 3000,the upper edge of the multipole range reported in this analysis.Further discussion of the point-source model is reserved forSection 6.1.

3.3. Cross Spectra

The next step in calculating the power spectrum is to cross-correlate single maps from different observations of the samefield. Each map is multiplied by the window for its field, zero-padded to the same size for all fields, then the Fourier transformof the map mA is calculated, where A is the observation index.The resulting Fourier-space maps have pixels of size δ� = 5on a side. We calculate the average cross spectrum between themaps of two observations A and B within an �-bin b:

DABb ≡

⟨�(� + 1)

2πH��Re[mA

�� mB∗�� ]

⟩�∈b

, (2)

where H�� is a two-dimensional weight array described in thenext paragraph, and �� is a vector in two-dimensional �-space.Each field typically has about 200 single maps in the inputset (see Section 3.1), resulting in ∼20, 000 cross spectra. Weaverage all cross spectra DAB

b for A = B to calculate a binnedpower spectrum Db for each field.

Given our observation strategy, the maps have statisticallyanisotropic noise; at fixed �, modes that oscillate perpendic-ular to the scan direction (�x = 0) are noisier than modesthat oscillate parallel to the scan direction. This anisotropicnoise—and the filtering we apply to reduce the noise (seeSection 2.2)—causes different modes in a given � bin to havedifferent noise properties. As in K11, we use a two-dimensionalweight H��, which accounts for the anisotropic noise in the maps.We define the weight array according to the following:

H�� ∝ (C th

� + N��

)−2, (3)

where C th� is the theoretical power spectrum used in simula-

tions described in Section 3.4.1, and N�� is the two-dimensionalcalibrated, beam-deconvolved noise power, which is calculatedfrom difference maps in which the right-going scans are sub-tracted from the left-going scans. The weight array is thensmoothed with a Gaussian kernel of width σ� = 450 to reducethe scatter in the noise power estimate, and then normalized tothe maximum value in each annulus. H�� is calculated indepen-dently for each observation field.

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3.4. Unbiased Spectra

The power Db is a biased estimate of the true sky power, Db,as a result of effects such as TOD filtering, projection effects,and mode mixing from the window. The biased and unbiasedestimates are related by the following:

Db′ ≡ Kb′bDb , (4)

where the K matrix accounts for the effects of the beams, TODfiltering, pixelization, windowing, and band-averaging. K canbe expanded as follows:

Kbb′ = Pb�

(M��′[W] F�′B2

�′)Q�′b′ . (5)

Q�′b′ is the binning operator and Pb� is its reciprocal (Hivonet al. 2002). The “mode-coupling matrix” M��′[W] accountsfor mixing modes between multipole moments that arises fromobserving a finite portion of the sky. We calculate M��′[W] ana-lytically from the window function W following the prescriptiondescribed by Lueker et al. (2010). Over the range of multipolesreported in this analysis, the elements of the mode-couplingmatrix depend only on the distance from the diagonal. F� is thetransfer function caused by TOD filtering and map pixelization,which is described in Section 3.4.1. B2

� is the beam functiondescribed in Section 2.3. For sufficiently large �-bins, the K ma-trix is invertible, allowing an unbiased estimate of the true skypower:

Db ≡ (K−1)bb′Db′ . (6)

3.4.1. Simulations and the Transfer Function

The transfer function F� is calculated from end-to-end sim-ulations. One hundred full-sky realizations are generated at aHealpix29 resolution of Nside = 8192. These simulated skiesinclude gravitationally lensed CMB anisotropy based on thebest-fit ΛCDM model from K11, a Poisson distribution of radiogalaxies, and Gaussian realizations of the thermal and kineticSZ effects and cosmic infrared background (CIB). The lensedrealizations of the CMB spectrum are generated out to � = 8000using LensPix (Lewis 2005). The Poisson radio-galaxy contri-bution is based on the De Zotti et al. (2010) model for sourcesbelow the 5σ detection threshold in the SPT-SZ survey, and theobserved counts (Vieira et al. 2010) above that flux. The shapeof the thermal SZ spectrum is taken from Shaw et al. (2010) withan amplitude taken from Reichardt et al. (2012). The kinetic SZspectrum is based on the fiducial model in Zahn et al. (2012).The CIB spectrum is taken from the best-fit values in Reichardtet al. (2012).

Unlike the simulations in K11, these simulations cover thefull sky. The full-sky simulations make it simple to account foroverlap between fields when calculating the sample varianceterm of the bandpower covariance matrix (see Section 3.5).These simulations also account for any effects caused byprojecting from the curved sky to flat-sky maps to first order inthe transfer function, although these effects should be negligible,as argued in K11.

These simulated skies are observed using the SPT pointinginformation and then filtered and processed into maps using thesame pipeline as for the real data. For each field, we calculatethe transfer function by comparing the average power spectrumof these simulated maps to the known input spectrum using aniterative scheme (Hivon et al. 2002).

29 http://healpix.jpl.nasa.gov

The transfer function is equal to ∼0.25 at � = 650 and reachesa plateau for � � 1200. The transfer function does not reachunity at any scale because of the strong filtering of �x � 300modes.

3.5. Bandpower Covariance Matrix

The bandpower covariance matrix quantifies the bin-to-bincovariance of the unbiased spectrum. The covariance matrixcontains signal and noise terms as well as terms accountingfor beam and calibration uncertainties. The signal term, oftenreferred to as “sample variance,” is calculated from the 100simulations described in Section 3.4.1. For each simulatedHealpix sky, we calculate the combined power spectrum fromall fields, then we measure the variance of these 100 estimates.This process naturally accounts for any overlap between fields.The noise term, or “noise variance,” is estimated directly fromthe data using the distribution of individual cross spectra DAB

b

as described by Lueker et al. (2010). The sample variance isdominant at multipoles below � � 2900. At smaller angularscales, the noise variance dominates.

The initial estimate of the bandpower covariance matrix haslow signal-to-noise on the off-diagonal elements. As in K11,we condition the covariance matrix to reduce the impact of thisuncertainty.

We must also account for the bin-to-bin covariance due tothe uncertainties in the beam function B�. We construct a beam-correlation matrix’ for each source of beam uncertainty:

ρρρbeamij =

(δDi

Di

)(δDj

Dj

), (7)

whereδDi

Di

= 1 −(

1 +δBi

Bi

)−2

. (8)

We sum these matrices to find the full beam-correlationmatrix, and we convert to a covariance matrix according tothe following:

Cbeamij = ρρρbeam

ij DiDj . (9)

3.6. Combining Fields

The analysis described in the previous sections produces 19sets of bandpowers and covariance matrices, one from eachfield. In either the limit of equal noise or the limit of samplevariance domination, the optimal weight for each field would beits effective area (i.e., the integral of its window). Because we areapproximately in these limits, we use area-based weights. Thus,the combined bandpowers and covariance matrix are given bythe following:

Db =∑

i

Dibw

i, (10)

Cbb′ =∑

i

Cibb′ (wi)2, (11)

where

wi = Ai∑i A

i(12)

is the area-based weight of the ith field. The area Ai is the sumof the window for the ith field.

We calculate the final covariance matrix as the sum ofthe signal plus noise covariance matrix, the beam-covariance

6

http://healpix.jpl.nasa.gov


matrix, and the calibration-covariance matrix. For the signaland noise terms, we combine the signal plus noise covariancematrices from all fields using Equation (11). We condition thiscombined covariance matrix using Equation (11) from K11.For the beam-covariance matrix, we take the beam-covariancematrices for each year (see Section 3.5) and combine them intoone composite beam-covariance matrix using the area-basedweight scheme. In this step, we account for the beam errorsthat are correlated between years. Last, we add the calibration-covariance matrix, defined as Ccal

ij = ε2DiDj , where ε = 0.026is the 2.6% uncertainty in the SPT power calibration discussedin Section 2.4.

3.7. Bandpower Window Functions

Bandpower window functions are necessary to compare themeasured bandpowers to a theoretical power spectrum. Thewindow function Wb

� /� is defined as follows:

C thb = (

Wb� /�

)C th

� . (13)

Following the formalism described in Section 3.4, we canwrite this as follows:

C thb = (K−1)bb′Pb′�′M�′�F�B

2� C

th� , (14)

which implies that30

Wb� /� = (K−1)bb′Pb′�′M�′�F�B

2� . (15)

We calculate the bandpower window functions to be used forthe final spectrum measurement as the weighted average of thebandpower window functions from each field.

4. TESTS FOR SYSTEMATIC ERRORS

It is important to verify that the data are unbiased bysystematic errors. We perform two types of tests: null tests andpipeline tests.

4.1. Null Tests

As is common in CMB analyses, we check for possiblesystematic errors by performing a suite of null tests that arefrequently referred to as jackknife tests. In each null test, allobservations are divided into two equally sized sets basedon a possible source of systematic error. Difference maps arethen calculated by subtracting the two sets, thus removing theastrophysical signal. The power spectrum of the difference mapsis calculated as described in the last section. This spectrum iscompared with an “expectation spectrum,” the power we expectto see in the absence of contamination from systematic errors.The expectation spectrum will generically be non-zero becauseof small differences in observation weights, filtering, etc., andis calculated by applying the null test to simulated maps. Theexpected power is small (D� < 2 μK2 at all multipoles) for alltests.

We perform the following six null tests.

30 Note: because of conventions in the CosmoMC package, the windowfunctions from the publicly downloadable “Newdat” files should be used asfollows:

Cthb =

(Wb

�

(� + 0.5)

(� + 1)

)Cth

� . (15)

1. Time. Observations are ordered by time, then dividedinto first- and second-half sets. This tests for long-termtemporally varying systematic effects.

2. Scan direction. Observations are divided into mapsmade from left- and right-going scans. This tests forscan-synchronous and scan-direction-dependent systematicerrors.

3. Azimuthal range. We split the data into observations takenat azimuths that we expect to be more or less susceptibleto ground pickup. These azimuth ranges are determinedfrom maps of the 2009 data made using ground-centered(azimuth/elevation) coordinates in which ground pickupadds coherently, as opposed to the usual sky-centered(R.A./decl.) coordinates. We use the ground-centered mapsthat were made for the analysis presented by K11. Althoughwe detect emission from the ground on large scales (� ∼ 50)in these ground-centered maps, this is not expected to biasour measurement; the amplitude of the ground pickup issignificantly lower on the smaller angular scales for whichthe bandpowers are being reported, and the observations fora given field are distributed randomly in azimuth. We usethe azimuth-based null test to verify this assertion.

4. Moon. Observations are divided into groups on the basis ofwhen the Moon was above and below the horizon.

5. Sun. Observations are divided into groups on the basis ofwhen the Sun was above and below the horizon. In this test,we include only the fields in which more than 25% of theobservations were taken with the Sun above the horizon.

6. Summed bolometer weights. We calculate the sum of allbolometer weights during each observation and order mapson the basis of this sum. This tests for bias introduced byincorrectly weighting observations or incomplete coveragein some maps.

For each test, the χ2 of the residual power is calculated relativeto the expectation spectrum in five bins with δ� = 500. Wecalculate the probability to exceed (PTE) this value of χ2 forfive degrees of freedom. All null tests had reasonable PTEs, aslisted in the next paragraph, with the exception of the AzimuthalRange null test, which produced a low PTE for the originalset of observations. This was interpreted as evidence for someground contamination. This interpretation was tested by cuttingseveral sets of 5% of the data, and recalculating the AzimuthalRange null test. Removing random 5% sets of the data did notchange the failure of the Azimuthal Range null test. However,cutting the 5% of the observations from each field with thehighest expected ground contamination resulted in passing theAzimuthal Range null test. Thus, this cut was included withthe other observation cuts, as described in Section 2.1. It isworth noting that the Azimuthal Range null test is the worst-case scenario for ground pickup; this null test systematicallyaligns azimuth ranges to maximize the ground contamination.In the analysis of the power spectrum of the sky signal, theground signal will add incoherently as the azimuth changes,thus reducing the power from ground contamination to a muchlower level than in this null test.

We find a flat distribution of PTEs ranging from 0.02 to0.99 for individual fields. The combined PTEs for the time,scan direction, azimuthal range, Moon, Sun, and summedbolometer weights are 0.26, 0.14, 0.17, 0.13, 0.30, and 0.63,respectively. It is important to note that these null tests areextremely conservative for the SPT power spectrum where theuncertainties are sample variance dominated over most of therange reported in this work. It is possible to have a failure in

7


500 1000 1500 2000 2500 3000

100

1000 SPT

500 1000 1500 2000 2500 30000

500

1000

1500

2000ACBARACTSPT, K11SPT, this work

Figure 3. Left panel: the SPT power spectrum. The leftmost peak at � ∼ 800 is the third acoustic peak. Right panel: a comparison of the new SPT bandpowers withother recent measurements of the CMB damping tail from ACBAR (Reichardt et al. 2009), ACT (Das et al. 2011b), and SPT (K11). Note that the point-source-maskingthreshold differs between these experiments, which can affect the power at the highest multipoles. To highlight the acoustic peak structure of the damping tail, we plotthe bandpowers in the right panel as �4C�/(2π ), as opposed to D� = �(� + 1)C�/(2π ) in the left panel. The solid line shows the theory spectrum for the ΛCDM model+ foregrounds that provides the best fit to the SPT+WMAP7 data. The bandpower errors shown in these plots contain sample and noise variance terms only; they donot include beam or calibration uncertainties.


these null tests without a significant impact on the final powerspectrum. The measured power D� in each null test was lessthan 2.2 μK2 in all bins.

Although we cannot perform a direct year-to-year null testbecause the data for any given field was taken within a singleyear, we have verified that the spectra from different years(and therefore different fields, including those used in K11)are consistent within the uncertainties of cosmic variance.

4.2. Pipeline Tests

We test the robustness of our pipeline with simulations. Inthese tests, we create simulated maps with an input spectrumthat differs from the ΛCDM model spectrum assumed in thecalculation of the transfer function. We then use our full pipelineto calculate the power spectrum of these simulated maps, andwe compare this spectrum with the input spectrum. We lookedat the following three categories of modifications.

1. A slope was added to the best-fit ΛCDM spectrum fromK11. This tests how well we can measure the slope of thedamping tail.

2. An additional Poisson point-source power term was added(see Section 6.1).

3. The input spectrum was shifted by δ� = 10. This tests howwell we can measure the locations of the acoustic peaks,and therefore θs .

In all cases, we recover the input spectrum to well within ouruncertainties.

We thus find no significant evidence for systematic contami-nation of SPT bandpowers.

5. BANDPOWERS

Following the analysis presented in Section 3, we measure theCMB temperature anisotropy power spectrum from 2540 deg2

of sky observed by the SPT between 2008 and 2011. We reportbandpowers in bins of δ� = 50 between 650 < � < 3000.The bandpowers and associated errors are listed in Table 2 andshown in Figures 3 and 4. The bandpowers, covariance matrix,

WMAP7

SPT

Figure 4. SPT bandpowers (blue), WMAP7 bandpowers (orange), and thelensed ΛCDM+foregrounds theory spectrum that provides the best fit to theSPT+WMAP7 data shown for the CMB-only component (dashed line), andthe CMB+foregrounds spectrum (solid line). As in Figure 3, the bandpowererrors shown in this plot do not include beam or calibration uncertainties.


and window functions are available for download on the SPTWeb site.31

These bandpowers clearly show the third to ninth acousticpeaks. As Figure 4 demonstrates, the anisotropy power mea-sured by this analysis (at 150 GHz, with a 50 mJy point-sourcecut) is dominated by primary CMB, with secondary anisotropyand foregrounds contributing significantly only at the highestmultipoles. These bandpowers provide the most precise mea-surement to date of the CMB power spectrum over the entiremultipole range presented in this analysis.

6. COSMOLOGICAL CONSTRAINTS

The SPT bandpowers are high signal-to-noise measurementsof the CMB temperature anisotropy over a large range of

31 http://pole.uchicago.edu/public/data/story12/

8

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Table 2SPT Bandpowers and Bandpower Errors

� Range �eff D� σ � Range �eff D� σ

(μK2) (μK2) (μK2) (μK2)

651–700 671 1786.2 59.5 1851–1900 1865 276.1 5.2701–750 720 1939.3 66.9 1901–1950 1915 238.2 4.5751–800 770 2426.4 67.2 1951–2000 1966 242.8 4.6801–850 820 2577.1 68.3 2001–2050 2015 245.8 4.9851–900 870 2162.3 53.8 2051–2100 2064 229.9 4.5901–950 920 1588.8 39.0 2101–2150 2114 194.2 3.8951–1000 969 1144.3 29.6 2151–2200 2164 170.6 3.41001–1050 1019 1068.0 27.2 2201–2250 2213 140.5 2.81051–1100 1069 1215.7 28.5 2251–2300 2265 135.0 2.61101–1150 1118 1193.8 29.1 2301–2350 2313 128.3 2.41151–1200 1169 1141.1 29.8 2351–2400 2364 124.8 2.71201–1250 1218 924.8 23.1 2401–2450 2413 115.8 2.21251–1300 1269 771.7 17.9 2451–2500 2462 100.7 2.21301–1350 1318 723.1 17.7 2501–2550 2512 96.7 2.31351–1400 1367 754.6 16.6 2551–2600 2562 83.3 2.01401–1450 1417 847.3 17.0 2601–2650 2613 85.7 1.81451–1500 1468 718.7 13.8 2651–2700 2663 83.9 1.91501–1550 1517 625.0 11.3 2701–2750 2712 76.4 1.81551–1600 1567 468.1 10.2 2751–2800 2761 71.7 1.81601–1650 1617 395.7 7.9 2801–2850 2811 62.9 1.71651–1700 1666 390.7 7.0 2851–2900 2860 57.6 1.61701–1750 1717 396.6 6.9 2901–2950 2910 57.6 1.61751–1800 1766 390.7 6.9 2951–3000 2961 56.6 1.61801–1850 1815 336.7 6.2

Notes. The �-band range, weighted multipole value �eff , bandpower D�, and associated bandpoweruncertainty σ of the SPT power spectrum. The errors are the square root of the diagonal elements ofthe covariance matrix, and do not include beam or calibration uncertainties.

angular scales and can be used to perform sensitive tests ofcosmological models. In this section, we present the constraintsthese bandpowers place on cosmological models. We firstconstrain the standard ΛCDM cosmological model. Next, weextend this model to constrain the amplitude of gravitationallensing of the CMB. We then consider models with free spatialcurvature and constrain the mean curvature of the observableuniverse. Last, we consider tensor perturbations and discuss theimplications of our observations for simple models of inflation.A wider range of cosmological models are tested in a companionpaper H12.

We parameterize the ΛCDM model with six parameters: thebaryon density Ωbh

2, the density of cold dark matter Ωch2, the

optical depth to reionization τ , the angular scale of the soundhorizon at last scattering θs , the amplitude of the primordialscalar fluctuations (at pivot scale k0 = 0.05 Mpc−1) Δ2

R , and thespectral index of the scalar fluctuations ns. With the exceptionof Section 6.8, we consider only flat-universe models where themean curvature of the observable universe Ωk = 0. In additionto the six parameters described earlier, we report several derivedparameters that are calculated from the six ΛCDM parameters.These are the dark-energy density ΩΛ, the Hubble constant H0in units of km s−1 Mpc−1, the current amplitude of linear-matterfluctuations σ8 on scales of 8 h−1 Mpc, the redshift of matter-radiation equality zEQ, and a hybrid-distance ratio reported byBAO experiments at two different redshifts rs/Dv(z = 0.35) andrs/Dv(z = 0.57), where rs is the comoving sound horizon sizeat the baryon drag epoch, DV (z) ≡ [(1 + z)2D2

A(z)cz/H (z)]1/3,DA(z) is the angular diameter distance, and H (z) is the Hubbleparameter.

6.1. Foreground Treatment

We marginalize over three foreground terms in all parameterfitting. The total foreground power, D

fg� , can be expressed as

follows:D

fg� = D

gal� + DSZ

� . (16)

These two terms represent the following.

1. Power from galaxies (Dgal� ), which can be subdivided into

a clustering term and a Poisson term. The Gaussian priorsused by K11 are applied to the amplitude of each term at� = 3000. For the clustering term, the prior is DCL

3000 =5.0 ± 2.5 μK2, based on measurements by Shirokoff et al.(2011). The angular dependence of the clustering term isDCL

� ∝ �0.8, which has been modified from that assumedby K11 to agree better with recent measurements (e.g.,Addison et al. 2012; Reichardt et al. 2013). For the Poissonterm, the prior is DPS

3000 = 19.3 ± 3.5 μK2. This is basedon the power from sources with S150 GHz < 6.4 mJy, asmeasured in Shirokoff et al. (2011), and the power fromsources with 6.4 mJy < S150 GHz < 50 mJy, as measured inVieira et al. (2010) and Marriage et al. (2011). The Poissonterm is constant in C� and thus varies as DPS

� ∝ �2.2. SZ power (DSZ

� ). The thermal and kinetic SZ effects areexpected to contribute to the observed CMB temperatureanisotropy. Both effects are expected to have similar powerspectrum shapes over the angular scales relevant to thisanalysis. Therefore, we adopt a single template to describeboth effects. The chosen template is the thermal SZ modelfrom Shaw et al. (2010). We set a Gaussian prior on the

9


amplitude of this term of DSZ3000 = 5.5 ± 3.0 μK2, as

measured in Shirokoff et al. (2011). This amplitude isdefined at 153 GHz, corresponding to the effective SPTband center.

We have tested that all parameter constraints are insensitive tothe details of the assumed foreground priors; we have completelyremoved the priors on the amplitudes of the foreground termsand recalculated the best-fit ΛCDM model, and find all ΛCDMparameters shift by less than 0.06σ . In addition, we see noevidence for significant correlations between the foregroundand cosmological parameters.

In the aforementioned model, we have not accounted forthe emission from cirrus-like dust clouds in the Milky Way.Repeating the calculation performed in K11, we cross-correlatethe SPT maps with predictions for the galactic dust emissionat 150 GHz in the SPT fields using model 8 of Finkbeineret al. (1999). We use this cross-correlation to estimate thepower from the galactic dust in the SPT fields and find thatit is small compared to the primary CMB power and the SPTbandpower errors. Specifically, subtracting this cirrus power canbe balanced by moving the foreground terms by amounts thatare small compared to their priors, so that the change in χ2 isless than 0.1σ . Thus, we conclude that galactic dust does notsignificantly contaminate the SPT power spectrum.

6.2. Estimating Cosmological Parameters

Our baseline model contains nine parameters: six for theprimary CMB and three for foregrounds. We explore thenine-dimensional parameter space using a Markov Chain MonteCarlo (MCMC) technique (Christensen et al. 2001) imple-mented in the CosmoMC32 (Lewis & Bridle 2002) softwarepackage. For reasons of speed, we use PICO33 (Fendt & Wandelt2007a, 2007b), trained with CAMB34 (Lewis et al. 2000), to cal-culate the CMB power spectrum. We have trained PICO for a10-parameter model that includes ΛCDM as well as several ex-tensions. We use PICO when working with any subset of thismodel space, and CAMB for all other extensions to ΛCDM.The effects of gravitational lensing on the power spectrum ofthe CMB are calculated using a cosmology-dependent lens-ing potential (Lewis & Challinor 2006). To sample the posteriorprobability distribution in regions of very low probability, we run“high-temperature” chains in which the true posterior, P, is re-placed in the Metropolis Hastings algorithm by PT = P 1/6. Thisallows the chain to sample the parameter space more broadly.We recover the correct posterior from the chain by importancesampling each sample with weight P/PT .

6.3. Goodness of Fit to the ΛCDM Model

We quantify the goodness of fit of the ΛCDM model to theSPT bandpowers by finding the spectrum that best fits the SPTbandpowers and calculating the reduced χ2 for the SPT data.The reduced χ2 for the SPT data is 45.9/39 (PTE = 0.21);thus, the ΛCDM model is a good fit to the SPT bandpowers.In H12, we consider several extensions to the ΛCDM model,and find that the data show some preference for several of thoseextensions.

32 http://cosmologist.info/cosmomc/33 https://sites.google.com/a/ucdavis.edu/pico34 http://camb.info/ (2012 January version)

6.4. External Datasets

In this work, we focus on parameter constraints from theCMB data, sometimes in conjunction with measurements ofthe Hubble constant (H0) or the BAO feature. For CMBmeasurements, we use the SPT bandpowers presented here aswell as the WMAP bandpowers presented in WMAP7. For H0measurements, we use the low-redshift measurement from Riesset al. (2011). For the BAO feature, we use a combination ofthree measurements at different redshifts: the WiggleZ surveycovering the redshift range 0.3 < z < 0.9 (Blake et al.2011), the Sloan Digital Sky Survey II survey (DR7) covering0.16 < z < 0.44 (Padmanabhan et al. 2012), and the BOSSsurvey covering 0.43 < z < 0.7 (Anderson et al. 2012).

Before combining the CMB, H0 , and BAO datasets, wecheck their relative consistency within the ΛCDM model.We quantify this consistency by calculating the χ2

min using areference dataset (e.g., CMB) and comparing it to the χ2

minobtained using a new dataset (e.g., CMB+H0). For example,χ2

min,[CMB+H0] − χ2min,[CMB] = 0.08. The PTE this Δχ2 given the

one new degree of freedom provided by the H0 measurementis 0.78, corresponding to an effective Gaussian significance of0.3σ . Using this metric, we find the following.

1. CMB and H0 differ by 0.3σ .2. CMB and BAO differ by 1.5σ .3. (CMB+BAO) and H0 differ by 1.8σ .4. (CMB+H0) and BAO differ by 2.1σ .

There is some tension between these datasets in the contextof the ΛCDM model. This could be evidence for a departurefrom ΛCDM, a systematic error in one or more of the datasets,or simply a statistical fluctuation. We assume the uncertaintiesreported for each of the datasets are correct and combine themto produce many of the results presented here.

6.5. SPT-only ΛCDM Constraints

We begin by examining parameter constraints from theSPT bandpowers alone. The SPT-only parameter constraintsprovide an independent test of ΛCDM cosmology and allow forconsistency checks between the SPT data and other datasets.Because the scalar amplitude Δ2

R and the optical depth τ arecompletely degenerate for the SPT bandpowers, we impose aWMAP7-based prior of τ = 0.088 ± 0.015 for the SPT-onlyconstraints.

We present the constraints on the ΛCDM model from SPT andWMAP7 data in Columns 2–4 of Table 3. As shown in Figure 5,the SPT bandpowers (including a prior on τ from WMAP7)constrain the ΛCDM parameters approximately as well asWMAP7. The SPT and WMAP7 parameter constraints areconsistent for all parameters; θs changes the most significantlyamong the five free ΛCDM parameters, moving by 1.5σ andtightening by a factor of 2.2 from WMAP7 to SPT. TheSPT bandpowers measure θs extremely well by virtue of thesheer number of acoustic peaks—seven—measured by the SPTbandpowers. The SPT constraint on ns is broader than theconstraint from WMAP7 because WMAP7 probes a much greaterdynamic range of angular scales. Degeneracies with ns degradethe SPT constraints on Δ2

R , the baryon density and, to a lesserextent, the dark-matter density.

6.6. Combined ΛCDM Constraints

Next, we present the constraints on the ΛCDM model fromthe combination of SPT and WMAP7 data. As previously

10

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https://sites.google.com/a/ucdavis.edu/pico

http://camb.info/


1.8 2.0 2.2 2.4 2.6 2.80.0

0.2

0.4

0.6

0.8

1.0

0.08 0.10 0.12

SPT+WMAP7WMAPSPT

1.03 1.04 1.05 0.6 0.7 0.8

0.06 0.08 0.10 0.12 0.140.0

0.2

0.4

0.6

0.8

1.0

0.85 0.90 0.95 1.00 1.9 2.0 2.1 2.2 2.3 2.4 65 70 75 80 85

Figure 5. One-dimensional marginalized likelihoods of the six parameters of the ΛCDM model, plus two derived parameters: the dark-energy density ΩΛ and theHubble constant H0. The constraints are shown for the SPT-only (blue dot-dashed lines), WMAP7-only (red dashed lines), and SPT+WMAP7 (black solid lines)datasets. With the exception of τ , the SPT bandpowers constrain the parameters approximately as well as WMAP7 alone. In particular, the SPT bandpowers measurethe angular sound horizon θs extremely well because they measure seven acoustic peaks. In the SPT-only constraints, the WMAP7 measurement of τ has been appliedas a prior; because of this, we do not plot an SPT-only line on the τ plot.


Table 3ΛCDM Parameter Constraints from the CMB and External Datasets

Parameter WMAP7 SPTa CMB CMB+H0 CMB+BAO CMB+H0+BAO(SPT+WMAP7)

Baseline parameters100 Ωbh

2 2.231 ± 0.055 2.30 ± 0.11 2.229 ± 0.037 2.233 ± 0.035 2.204 ± 0.034 2.214 ± 0.034Ωch

2 0.1128 ± 0.0056 0.1056 ± 0.0072 0.1093 ± 0.0040 0.1083 ± 0.0033 0.1169 ± 0.0020 0.1159 ± 0.0019109Δ2

R 2.197 ± 0.077 2.164 ± 0.097 2.142 ± 0.061 2.138 ± 0.062 2.161 ± 0.057 2.160 ± 0.057ns 0.967 ± 0.014 0.926 ± 0.029 0.9623 ± 0.0097 0.9638 ± 0.0090 0.9515 ± 0.0082 0.9538 ± 0.0081100 θs 1.0396 ± 0.0027 1.0441 ± 0.0012 1.0429 ± 0.0010 1.0430 ± 0.0010 1.04215 ± 0.00098 1.04236 ± 0.00097τ 0.087 ± 0.015 0.087 ± 0.015 0.083 ± 0.014 0.084 ± 0.014 0.076 ± 0.012 0.077 ± 0.013

Derived parametersb

ΩΛ 0.724 ± 0.029 0.772 ± 0.033 0.750 ± 0.020 0.755 ± 0.016 0.709 ± 0.011 0.7152 ± 0.0098H0 70.0 ± 2.4 75.0 ± 3.5 72.5 ± 1.9 73.0 ± 1.5 69.11 ± 0.85 69.62 ± 0.79σ8 0.819 ± 0.031 0.772 ± 0.035 0.795 ± 0.022 0.791 ± 0.019 0.827 ± 0.015 0.823 ± 0.015zEQ 3230 ± 130 3080 ± 170 3146 ± 95 3124 ± 78 3323 ± 50 3301 ± 47100 rs

DV(z = 0.35) 11.43 ± 0.37 12.15 ± 0.55 11.81 ± 0.29 11.89 ± 0.24 11.28 ± 0.12 11.35 ± 0.12

100 rsDV

(z = 0.57) 7.58 ± 0.21 7.98 ± 0.31 7.80 ± 0.16 7.84 ± 0.13 7.505 ± 0.068 7.545 ± 0.065

Notes. The constraints on cosmological parameters from the ΛCDM model, given five different combinations of datasets. We report the median of the likelihooddistribution and the symmetric 68.3% confidence interval about the mean.a We impose a WMAP7-based prior of τ = 0.088 ± 0.015 for the SPT-only constraints.b Derived parameters are calculated from the baseline parameters in CosmoMC. They are defined at the end of Section 6.1.

mentioned, we will refer to the joint SPT+WMAP7 likelihood asthe CMB likelihood. We then extend the discussion to includeconstraints from CMB data in combination with BAO and/orH0 data.

We present the CMB constraints on the six ΛCDM parametersin the fourth column of Table 3. Adding SPT bandpowers to theWMAP7 data tightens these parameter constraints considerablyrelative to WMAP7 alone. Of these parameters, the constraint onθs sees the largest improvement; adding SPT data decreases theuncertainty on θs by a factor of 2.70 relative to WMAP7 alone.

Constraints on Ωbh2, Ωch

2, and ΩΛ tighten by factors of 1.49,1.40, and 1.45, respectively. For comparison, the addition of theK11 bandpowers to WMAP7 led to improvements of 1.33, 1.16,and 1.11, respectively. Last, the constraint on the scalar spectralindex tightens by a factor of 1.44 to give ns < 1.0 at 3.9σ .

The preferred values for Ωbh2 and Ωch

2 for WMAP7 donot shift significantly with the addition of the SPT data and,therefore, neither does the sound horizon, rs, which dependsonly on these parameters in the ΛCDM model. Thus, the shiftin θs = rs/DA, driven by the SPT acoustic peak locations,

11


must lead to a shift in DA. Shifting DA requires shifting ΩΛ(or, equivalently, H0). This shift in ΩΛ and H0 can be seen inFigure 5.

We explore how the SPT+WMAP7 constraints on the ΛCDMmodel change if different �-ranges of the SPT data are used inAppendix A.

We show the parameter constraints after adding the H0and/or BAO data to the CMB data in the last three columnsof Table 3. Combining the CMB bandpowers with thislow-redshift information tightens the constraints on Ωch

2 andΩΛ by a further factor of 1.2 or 2 for CMB+H0 and CMB+BAO,respectively, with smaller but significant improvements to otherparameters. Of special note is the constraint on the scalar spec-tral index, which tightens to ns = 0.9638 ± 0.0090 for theCMB+H0 dataset, 0.9515 ± 0.0082 for the CMB+BAO dataset,and 0.9538 ± 0.0081 for the CMB+H0+BAO dataset. Theseconstraints correspond to a preference for ns < 1 at 4.0σ , 6.1σ ,and 5.9σ respectively, for these three data combinations. Thisis the most significant reported measurement of ns = 1 to date.See Section 6.9 for a more detailed discussion of constraintson ns.

6.6.1. Consistency of ΛCDM Constraints

Comparing the best-fit cosmological model with K11, allΛCDM parameters are consistent at <1σ with the exception ofθs , which shifts up by 1.0σ . It is not surprising that the mostsignificant shift is seen in θs ; of the six ΛCDM parameters, SPTdata has the strongest effect on the θs constraint, as can be seenin Figure 5. Thus, the results in this paper are consistent withthose from K11.

The σ8 constraints presented here are consistent with previousmeasurements. The SPT-only and WMAP7-only values areconsistent at ∼1σ , with the SPT data preferring a lower value.The ACT+WMAP7 constraint, 0.813 ± 0.028 (Dunkley et al.2011), is consistent with that from SPT+WMAP7, although wenote that the WMAP7 data are used in both. Comparing to X-raymeasurements of cluster abundance, we rescale the σ8 constraintfrom Vikhlinin et al. (2009) to the SPT+WMAP7 value of ΩM

to find σ8 = 0.813 ± 0.027, which is again consistent with ourmeasured values. Optical and SZ-based surveys give comparableand consistent constraints, for example, Rozo et al. (2010)and Reichardt et al. (2013). Last, SPT gravitational lensingmeasurements are consistent; van Engelen et al. (2012) foundσ8 = 0.810 ± 0.026 (WMAP7+SPTLensing). Further discussionof σ8 constraints, particularly in the context of the ΛCDM +Σmν

model, can be found in H12.

6.7. Gravitational Lensing

As CMB photons travel from the surface of last scattering tothe Earth, their paths are deflected by gravitational interactionswith intervening matter. This gravitational lensing encodesinformation about the distribution of matter along the line ofsight, providing a probe of the distance scale and growth ofstructure at intermediate redshifts (0.5 � z � 4). Lensingdistorts the CMB anisotropy by shifting the apparent position ofCMB photons on the sky, with typical deflection angles of 2.′5,which are coherent over degree scales. This process mixes powerbetween multipoles in the CMB temperature power spectrum,which smooths the acoustic peak structure and increases thepower in the damping tail at small angular scales (see Lewis &Challinor 2006 for a review).

The ΛCDM model already includes the effects of gravita-tional lensing. To quantify the sensitivity of the SPT bandpowers

to gravitational lensing, we extend the ΛCDM model to includeone additional free parameter, AL (Calabrese et al. 2008), whichrescales the lensing potential power spectrum, C

φφ

� , accordingto the following:

Cφφ

� → ALCφφ

� . (17)

We recalculate Cφφ

� in a cosmology-dependent manner at eachpoint in the MCMC. In effect, the AL parameter modulates theamplitude of gravitational lensing. Setting AL = 1 correspondsto the standard theoretical prediction and recovers the standardΛCDM model, while setting AL = 0 corresponds to nogravitational lensing. In the parameter fits, the range of AL isallowed to extend well above 1 and below 0.

The first detection of gravitational lensing in the CMB usedlensing-galaxy cross-correlations (Smith et al. 2007; Hirata et al.2008), and subsequent papers using this technique have achievedhigher signal-to-noise detections (Bleem et al. 2012; Sherwinet al. 2012).

The impact of lensing on the CMB power spectrum has beendetected in combinations of WMAP with ACBAR (Reichardtet al. 2009), WMAP with ACT (Das et al. 2011a), and WMAPwith SPT (K11). Using this effect, Das et al. (2011a) foundAL = 1.3+0.5

−0.5 at 68% confidence. K11 found that the constrainton A0.65 had the most Gaussian shape and thus reportedA0.65

L = 0.94 ± 0.15, a ∼5σ detection of lensing.CMB lensing has also been detected through the CMB

temperature four-point function in ACT (Das et al. 2011b) andSPT (van Engelen et al. 2012) data. Das et al. (2011b) usedACT data to measure35AML

L = 1.16±0.29. In van Engelen et al.(2012), the SPT four-point analysis was applied to a subset ofthe data used in this work to measure AL = 0.90 ± 0.19, whichwas previously the most significant detection of CMB lensingto date, ruling out no lensing at 6.3σ .

We determine the significance of the observed CMB lensingby constraining AL with the measured CMB power spectrum.The one-dimensional likelihood function for AL is shown inFigure 6. The significance of the detection is quantified bycalculating the probability for AL � 0, P (AL � 0). AsAL = 0 is far out in the tail of the likelihood distributionof this parameter, we use high-temperature MCMC’s chainsto estimate P (AL � 0). Using SPT data only, we measureP {SPT}(AL � 0) < 1.3 × 10−9, the equivalent of a 5.9σpreference for AL > 0 in a Gaussian distribution. For SPT+WMAP7 we measure the following:

P {CMB}(AL � 0) � 2.4 × 10−16 , (18)

which corresponds to a 8.1σ detection of lensing in a Gaussiandistribution.

Next, we report constraints on AL. Using SPT+WMAP7 datawe find the following:

AL = 0.86+0.15(+0.30)−0.13(−0.25) , (19)

where asymmetric 1σ (68.3%) and 2σ (95.5%) errors areshown. The observed lensing amplitude is consistent at 1σ withtheoretical predictions in the ΛCDM model.

35 In Das et al. (2011b), AL is calculated as the best-fit amplitude to the lensingpotential in the cosmological model with the maximum likelihood ΛCDMparameters. This is in contrast with what was done for the CMB temperaturepower-spectrum measurements of AL, where constraints on AL have beenmarginalized over cosmological parameters. The corresponding maximumlikelihood measure from van Engelen et al. (2012) is AML

L = 0.86 ± 0.16.

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Figure 6. SPT bandpowers allow a significant detection of gravitationallensing through the effective smoothing of the acoustic peaks. Here, weshow the one-dimensional likelihood function for AL, a rescaling parameterfor the gravitational lensing potential power spectrum (Cφφ

� → ALCφφ� ). The

SPT+WMAP7 data lead to a 8.1σ detection of CMB lensing, the most significantdetection to date.


6.8. Mean Curvature of the Observable Universe

The low-redshift information imprinted on the CMB bygravitational lensing, along with the other information in theCMB anisotropy power spectrum, enables the placement oftight constraints on the mean curvature of the observableuniverse. The magnitude of the mean curvature today can beparameterized by Ωk ≡ −K/H 2

0 where√

1/|K| is the lengthscale over which departures from Euclidean geometry becomeimportant. Inflationary models generically predict |Ωk| � 10−5

(e.g., Knox 2006); thus, a significant measurement of Ωk = 0would challenge our standard picture of the very early universe.A positive value for Ωk(K < 0) could be obtained by thenucleation of a bubble of lower vacuum energy in a surroundingmedium with higher vacuum energy followed by a short periodof inflation (Bucher et al. 1995). A determination that Ωk

is negative with high statistical significance would be veryinteresting; such a detection would be difficult to understandin the theoretical framework of inflation, challenge the stringtheory landscape picture, and rule out the de Sitter equilibriumcosmology of Albrecht (2011).

Absent lensing effects, one can leave the CMB power spec-trum unchanged while simultaneously varying Ωk and ΩΛ ina way that keeps the distance to last scattering fixed (Bondet al. 1997; Zaldarriaga et al. 1997).36 Historically, the CMBdata placed very coarse constraints on Ωk , with finer constraintsonly possible with the addition of other data sensitive to Ωk andΩΛ such as H0 measurements and determinations of Ωm (e.g.,Dodelson & Knox 2000) from, for example, the baryon fractionin clusters of galaxies (White et al. 1993).

The sensitivity of the CMB to low-redshift informationthrough gravitational lensing makes it possible to constrainthe mean curvature of the observable universe, and thus thecosmological constant, using the CMB alone. The lensingamplitude is sensitive to the distance and growth of structure at

36 There is an exception to this at very large scales because of the lateIntegrated Sachs-Wolfe effect, but sample variance makes these changesunobservably small in the CMB power spectrum.

intermediate redshifts (0.5 � z � 4). These observables are, inturn, sensitive to curvature, dark energy, and neutrino masses, asdiscussed in H12. The recent detections of CMB lensing havemeasured an amplitude that is consistent with ΩK ∼ 0 andΩΛ ∼ 0.7 (Das et al. 2011a, 2011b; Sherwin et al. 2011; Keisleret al. 2011; van Engelen et al. 2012). Simply put, the strengthof CMB lensing in a universe with no dark energy and positivemean curvature would be much larger than that observed (see,e.g., Sherwin et al. 2011).

Using the SPT+WMAP7 bandpowers, we measure the meancurvature of the observable universe using only the CMB:

Ωk = −0.003+0.014−0.018 . (20)

This result tightens curvature constraints over WMAP7 com-bined with low-redshift probes by ∼20%. This constraint isconsistent with zero mean curvature, and corresponds to adark-energy density of ΩΛ = 0.740+0.045

−0.054 and Hubble constant,H0 = 70.9+9.2

−8.0 (km s−1 Mpc−1). This measurement rules outΩΛ = 0 at 5.4σ using the CMB alone. The right panel ofFigure 7 shows the corresponding two-dimensional marginal-ized constraints on ΩM and ΩΛ. We have confirmed that thestrength of this constraint relies on the lensing signal; allow-ing AL to vary freely causes the curvature constraint to degradedramatically.

CMB lensing enables an independent constraint on curvature,although the most powerful curvature constraints still comefrom combining CMB data with other low-redshift probes (e.g.,H0 , BAO). The curvature constraint using CMB+H0 data isΩk = 0.0018 ± 0.0048, while the constraint using CMB+BAOdata is Ωk = −0.0089 ± 0.0043. The tightest constraint onthe mean curvature that we consider comes from combining theCMB, H0 , and BAO datasets:

Ωk = −0.0059 ± 0.0040. (21)

While the CMB+BAO constraint shows a 2.0σ preference forΩk < 0, the significance of this preference decreases as moredata are added. The tightest constraint, coming from CMB+H0+BAO, is consistent with zero mean curvature at 1.5σ . Theseresults are summarized in Figure 7. As discussed by H12, otherextensions of ΛCDM can also explain the data (e.g., allowingfor both non-zero mean curvature and a dark-energy equation ofstate w = −1); thus, these constraints are significantly degradedwhen considering multiple extensions to ΛCDM.

6.9. Inflation

Cosmic inflation is an accelerated expansion in the earlyuniverse (Guth 1981; Linde 1982; Albrecht & Steinhardt 1982)that generically leads to a universe with nearly zero meancurvature and a nearly scale-invariant spectrum of “initial”density perturbations (Mukhanov & Chibisov 1981; Hawking1982; Starobinsky 1982; Guth & Pi 1982; Bardeen et al.1983) that evolved to produce the observed spectrum of CMBanisotropies. Models of inflation compatible with current datagenerally predict, over the range of observable scales, scalarand tensor perturbations well characterized by a power law inwavenumber k,

Δ2R(k) = Δ2

R(k0)

(k

k0

)ns−1

, (22)

Δ2h(k) = Δ2

h(k0)

(k

k0

)nt

. (23)

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Figure 7. Low-redshift information imprinted on the CMB by gravitational lensing, along with the other information in the CMB anisotropy power spectrum, enablesthe placement of tight constraints on the mean curvature of the observable universe. The addition of low-redshift probes further tighten CMB-only constraints onthe mean curvature. Left panel: the one-dimensional marginalized constraints on Ωk from SPT+WMAP7 (black solid line), SPT+WMAP7+H0 (orange dashed line),SPT+WMAP7+BAO (black dotted line), and SPT+WMAP7+BAO+H0 (blue dot-dashed line). The SPT+WMAP7 datasets measure the mean curvature of the observableuniverse to a precision of ∼1.5%, while combining SPT,WMAP7, and either H0 or BAO data reduces the uncertainty by a factor of ∼3. Right panel: the two-dimensionalconstraints on ΩM and ΩΛ from the SPT+WMAP7 data alone. The SPT+WMAP7 data rule out ΩΛ = 0 at 5.4σ .


Here, Δ2R(k0) is the amplitude of scalar (density) perturbations

specified at pivot scale k = k0, with scale dependence controlledby the index ns, while Δ2

h(k0) is the amplitude of tensor(gravitational wave) perturbations specified at the same pivotscale, with scale dependence set by nt. The amplitude of thetensor perturbation spectrum is expressed in terms of the tensor-to-scalar ratio:

r = Δ2h(k)

Δ2R(k)

∣∣∣∣k=0.002 Mpc−1

. (24)

For single-field models in slow-roll inflation, nt and r are relatedby a consistency equation (Copeland et al. 1993; Kinney et al.2008):

nt = −r/8. (25)

The tensor and scalar perturbations predicted by such models ofinflation can thus be characterized by the three parameters ns,Δ2

R(k0), and r.In the following, we first consider constraints on ns assuming

r = 0, and then on both ns and r. We then compare theconstraints in the ns–r plane to predictions of inflationarymodels. Constraints on the scale dependence of the spectralindex (dns/d ln k) are considered in the companion paper H12.

6.9.1. Constraints on the Scalar Spectral Index

Inflation is a nearly time-translation invariant state; however,this invariance must be broken for inflation to eventually cometo an end. The wavelength of perturbations depends solelyon the time that they were produced; thus, a time-translationinvariant universe would produce scale-invariant perturbations(ns = 1).37 The prediction that inflation should be nearly, butnot fully, time-translation invariant gives rise to the predictionthat ns should deviate slightly from unity (Dodelson et al. 1997).

Because of the special status of ns = 1, and because we gen-erally expect a departure from ns = 1 for inflationary models,detecting this departure is of great interest. K11 combined data

37 Scale invariance here means that the contribution to the rms densityfluctuation from a logarithmic interval in k, at the time when k = aH , isindependent of k. Here, a(t) is the scale factor and H ≡ a/a is the Hubbleparameter.

from SPT and WMAP7 to measure a 3.0σ preference for ns < 1in a ΛCDM model, with ns = 0.966 ± 0.011. We show ourconstraints on ns for the ΛCDM model from several combineddatasets in the left panel of Figure 8. All datasets strongly preferns < 1.

Using SPT+WMAP7 data, we find the following:

ns = 0.9623 ± 0.0097. (26)

For this dataset, we find P (ns > 1) = 4×10−5, a 3.9σ departurefrom ns = 1; ns < 1 is heavily favored.

Including BAO data substantially shifts and tightens theconstraints on ns, as can be seen in Figure 8. The BAO distancemeasure rs/DV depends on ΩΛ, breaking the partial degeneracybetween ΩΛ and ns in the CMB power spectrum. The BAOpreference for lower ΩΛ pulls the central value of ns down tons = 0.9515 ± 0.0082. Using a high-temperature MCMC, wemeasure the probability for ns to exceed one to be 1.1 × 10−9,corresponding to a 6.0σ detection of ns < 1.

Including H0 data has a smaller effect on the ns constraintthan BAO, slightly disfavoring low-ns values as seen in Figure 8.The mechanism for the improvement is the same as for BAO,however, the CMB and H0 datasets individually prefer similarvalues of ΩΛ. Thus, the two datasets tighten the ns constraintaround the CMB-only value, leading to ns = 0.9638 ±0.0090. Using the combined CMB+H0 dataset, we measure theprobability for ns to exceed one to be 3.1×10−5, correspondingto a 4.0σ preference for ns < 1.

As expected, combining CMB with both BAO and H0 datanudges the constraint on ns up slightly from the CMB+BAOconstraint to ns = 0.9538 ± 0.0081, thus weakening thepreference for ns < 1 slightly from 6.0 to 5.7σ .

In summary, regardless of which datasets we use, the datastrongly prefer ns < 1 in the ΛCDM model.

The importance of detecting a departure from scale invarianceleads us to review our modeling assumptions. Specifically,are there extensions to the standard ΛCDM model that couldreconcile the data with a scale-invariant spectrum, ns = 1? Weanswer this question by calculating the ns constraints from theCMB+H0+BAO dataset for several physically motivated ΛCDMmodel extensions.

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Figure 8. Data strongly prefer departures from a scale-invariant primordial power spectrum (ns < 1), as predicted by inflation. Left panel: the marginalizedone-dimensional constraints on ns for the standard ΛCDM model (with r = 0) using several datasets. SPT data tightens the constraint on ns relative to WMAP7 alone.Adding BAO data further tightens this constraint and leads to a preference for lower values of ns, while adding H0 has little effect. Right panel: the one-dimensionalmarginalized constraints on ns from the SPT+WMAP7+H0+BAO dataset given three different models. Plotted are ΛCDM (black solid line), ΛCDM+Σmν (greendot-dashed line) as a typical case for extensions affecting the late-time universe, and ΛCDM+Neff (purple dashed line) as a typical case for extensions affecting theSilk damping scale. Of the extensions considered here, only those that affect the damping tail—in this case, by varying neutrino species—causes noticeable movementtoward ns = 1. We note that in all cases the data robustly prefer a scale-dependent spectrum with ns < 1.


We consider two classes of model extensions: those thatcan affect the slope of the CMB damping tail and those thatcannot. As a representative case of the first class of extensions,we consider ΛCDM+Neff , in which the number of relativisticspecies is allowed to vary. As an example of the second classof extensions, we consider massive neutrinos ΛCDM+Σmν

(with Neff fixed at its fiducial value of 3.046). These exampleextensions as well as several others are explored in considerabledetail in H12.

Of the extensions considered, only models that can affect theslope of the damping tail significantly increase the likelihood ofns = 1. The results of this test are displayed in the right panelof Figure 8, where we show the marginalized constraints on nsfrom the CMB+BAO+H0 dataset. Even the Neff extension doesa poor job reconciling the data with a scale-invariant spectrum;the cumulative probability for ns > 1 is 6.1×10−3, constrainingns to be less than one at 2.5σ . The preferred value for Neff is fargreater than the nominal value of 3.046 in the limited allowedparameter space.

We conclude that the data robustly prefer a scale-dependentspectrum with ns < 1.

6.9.2. Constraints on Tensor Perturbations

The last inflationary parameter we consider is the tensor-to-scalar ratio r. Because r is related to the energy scaleof inflation,38 a detection of r would provide an extremelyinteresting window onto the early universe. We first considerthe marginalized constraints on r for the ΛCDM+r model,shown in Table 4 and Figure 9, then move on to a comparisonwith inflationary models in the ns–r plane in Section 6.9.3 andFigure 10.

One can think of the r measurement in the following way.The CMB power spectrum is first measured at � � 60 (wheretensor perturbations are negligible) to determine the ΛCDM

38 The tensor-to-scalar ratio r is proportional to the inflaton potential V (φ) andthe energy scale of inflation is proportional to V (φ)1/4 (Kinney 2003;Baumann et al. 2009).

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

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r

WMAP7WMAP7+BAOSPT+WMAP7SPT+WMAP7+BAO

Figure 9. This figure highlights the contributions of the SPT data to constraintson the tensor-to-scalar ratio, r. We show four datasets: WMAP7 (red dashedline), WMAP7+BAO (orange dot-dashed line), SPT+WMAP7 (black solid line),and SPT+WMAP7+BAO (black dotted line). Note that the WMAP7+BAO andSPT+WMAP7 likelihood functions are nearly identical. SPT data tightens ther constraint significantly, regardless of whether BAO data are included. Whileadding low-redshift H0 measurements has minimal effect on the constraints onr (not shown), adding low-redshift information from BAO tightens constraintson r considerably. SPT+WMAP7 constrains r < 0.18 (95% C.L.), while addinglow-redshift BAO measurements tightens the constraint to r < 0.11 (95% C.L.).


model parameters, thus determining the scalar contributionsto the power spectrum. This scalar-only spectrum is thenextrapolated to low �; any excess power observed is dueto tensor perturbations. In this way, although the SPT datapresented here do not directly measure power that could be fromgravitational waves, by pinning down other model parameters,the extrapolation of the scalar power to large scales is moreprecise.

Even with the parameters that determine the scalar powerspectrum perfectly known, there would still be significant

15


Table 4Constraints on ns and r from CMB and External Datasets

Model Parameter CMB CMB+H0 CMB+BAO CMB+H0+BAO(SPT+WMAP7)

ΛCDM ns 0.9623 ± 0.0097 0.9638 ± 0.0090 0.9515 ± 0.0082 0.9538 ± 0.0081ΛCDM+r ns 0.969 ± 0.011 0.9702 ± 0.0097 0.9553 ± 0.0084 0.9577 ± 0.0084

r (95% C.L.) <0.18 <0.18 <0.11 <0.11

Figure 10. We compare the constraints on the ΛCDM+r model with predictionsfrom models of inflation in the ns–r plane. We show the two-dimensionalconstraints on r and ns as colored contours at the 68% and 95% confidencelevels for three datasets: WMAP7 (gray contours), CMB (red contours), andCMB+H0+BAO (blue contours). Adding the SPT bandpowers partially breaksthe degeneracy between ns and r in the WMAP7 constraint, which can beseen clearly moving between the gray and red contours. Plotted over theconstraint contours are predictions for several models of inflation. We restrictour comparison with model predictions to the simplest cases of slow-rollinflation mediated by a single scalar field as reviewed in Baumann et al.(2009). Solid black line: the predictions of exponential inflation (V (φ) ∝exp[

√16πφ2/(p M2

Pl)]) lie on this line. In exponential inflation, increasingp moves the prediction toward the Harrison–Zel’dovich–Peebles point ns = 1,r = 0. Black lines with colored circles: the predictions of power-law potentialinflation models (V (φ) ∝ (φ/μ)p , p > 0) for five different values of p lie onthe corresponding line. The predictions in the r–ns plane are a function of N,where N is expected to be in the range N ∈ [50, 60]. Purple region: this regionrepresents the upper limit on r from large-field hilltop inflation models. Thewidth of the curve represents the uncertainty because of varying N ∈ [50, 60],but it does not take into account effects of higher order terms in the potentialthat may become important at the end of inflation.


uncertainty in the value of r because of cosmic variance. Knox& Turner (1994) showed that for different realizations of auniverse where r = 0, 50% of cosmic variance-limited fullsky temperature surveys will be able to place a limit of r < 0.1at 95% confidence. As we will see, the results we present hereapproach that limit.

WMAP7 data alone have been used to constrain r < 0.36at 95% confidence in the ΛCDM+r model (Larson et al.2011). Prior to the current SPT analysis, the tightest publishedconstraint on r was reported by Sanchez et al. (2012), who usedthe combination of datasets from BOSS-CMASS, WMAP7, andK11 to constrain r < 0.16 at 95% confidence.

The bandpowers presented here lead to a significant reductionin the upper limit on r. These measurements are summarized inTable 4. As shown in Figure 9, SPT bandpowers tighten the

constraint on r regardless of whether low-redshift informationfrom BAO or H0 is included. Using the CMB datasets, wemeasure the following:

r < 0.18 (95% C.L.). (27)

This limit remains unchanged for the CMB+H0 datasets, whilefrom the CMB+BAO datasets we measure r < 0.11 (95%C.L.). As with constraints on ns, the improvement with theaddition of BAO comes from the improved BAO constraints onΩΛ breaking a partial three-way degeneracy between ΩΛ, ns,and r. In contrast, adding in the H0 measurement to CMB datadoes not significantly tighten the CMB-only result because theH0 measurement sharpens up the ΩΛ distribution around thevalues that allow for larger r. It is worth noting that the SPTbandpowers tighten the constraint on r even in the presence ofBAO data; this can be seen in the tighter r constraint from theWMAP7+BAO to the SPT+WMAP7+BAO datasets in Figure 9.

Using the combination of the CMB+BAO+H0 datasets, wemeasure the following:

r < 0.11 (95% C.L.); (28)

this limit is unchanged from the CMB+BAO constraint. The H0and BAO data pull in opposite directions on the CMB-only datain the rs/DV –H0 plane; however, the statistical weight of theBAO measurements dominate the constraint in the combineddataset.

With the addition of low-redshift information from the BAOmeasurement, the constraints on tensor perturbations approachthe theoretical limit of what can be achieved with temperatureanisotropy alone. With the combination of these data, thecosmological parameters controlling the scalar perturbationsare now sufficiently well known that they do not significantlydegrade the limit on r. This is evident in Figure 10, where addingBAO removes most of the degeneracy between ns and r. Furtherimprovements now await precision B-mode CMB polarizationobservations (Seljak & Zaldarriaga 1997; Kamionkowski et al.1997). The lowest upper limit on r from B-modes is currentlyr < 0.7 (95% C.L.) from the BICEP experiment (Chiang et al.2010).

6.9.3. Implications for Models of Inflation

We now turn to a comparison of model predictions with dataconstraints in the ns–r plane. This comparison is illustratedin Figure 10, where we show the two-dimensional marginal-ized constraints from three combinations of data with predic-tions from simple models of inflation over-plotted. First, wenote that the confidence contours for the CMB-only case inFigure 10 show the expected positive correlation between nsand r. Essentially, the suppression of large-scale power when in-creasing ns can be countered by adding extra large-scale powersourced by tensors. The SPT data disfavor large values of ns(and hence r), significantly reducing the degeneracy betweenthese two parameters. With BAO data added to SPT+WMAP7,

16


the ns–r correlation nearly disappears. Adding H0 data has littleeffect on constraints from the CMB or CMB+BAO datasets, re-moving only the smallest allowed values of ns in both cases. Asmentioned earlier, we are approaching the cosmic-variance limitfor the temperature anisotropy on measuring r—at which pointimproved knowledge of the six ΛCDM parameters no longertranslates into better limits on r.

We restrict the model comparisons to the simplest cases ofsingle-field, slow-roll inflation, as reviewed in Baumann et al.(2009). Models can be broadly characterized according to howmuch the inflaton field φ changes from the time perturbationson observably large scales were being produced until the endof inflation; this change in φ defined at Δφ. Models in whichΔφ is larger than the Planck mass (MPl) are classified as “large-field” models, while those in which Δφ < MPl are classified as“small-field” models. The dividing line between the two casescorresponds to r = 0.01.

Here, we examine large-field inflation models, consideringseveral forms of the inflaton effective potential: large-fieldpower-law potential inflation models (V (φ) ∝ (φ/μ)p, p > 0),large-field hilltop inflation models (V (φ) ∝ 1 − (φ/μ)2), andexponential inflation models (V (φ) ∝ exp[

√16πφ2/(p M2

Pl)]).Large-field power-law potential models have the fewest free

parameters, and we discuss them first. Given a choice of p,these models have only one free parameter, and this parameter ishighly constrained by the requirement of reproducing the well-known amplitude of the scalar perturbation spectrum. Thus,these models make fairly localized predictions in the ns–r plane.The uncertainty in these predictions is dominated by the detailsof the end of inflation, which are not specified by V (φ) but,instead, depend on the coupling of the inflaton field φ to otherfields. This uncertainty can be captured by the parameter Nwhere eN gives the increase in the scale factor between the timewhen the observable scale leaves the horizon and the end ofinflation.39 Assuming a standard slow-roll inflation scenario,40Nis expected to lie in the range 50–60 (Liddle & Leach 2003). Thespread in values is dominated by uncertainty in how much theenergy density drops between the end of inflation and reheating,although this range can be extended in either direction bymodifications to the standard thermal history. In Figure 10 weconsider several large-field power-law potential models, eachwith a different value of p, and indicate the predictions of eachmodel as N varies between 50 and 60.

The λφ4 (p = 4) and λφ3 (p = 3) models were ruled out byearlier data (Komatsu et al. 2009; Dunkley et al. 2011), giventhe expected range of N. These models possibly could havebeen saved with a non-standard post-inflation thermal history(designed to make N very large, thus moving the predictiontoward r = 0 and ns = 1 in Figure 10), but such a maneuver nolonger works. The CMB+BAO dataset excludes the λφ4 modelwith greater than 95% confidence for all values of N, while theφ3 (p = 3) model is excluded with greater than 95% confidenceby the CMB or CMB+H0 datasets given the expected rangeof N, and excluded with greater than 95% confidence by theCMB+BAO datasets regardless of N.

39 A note of clarification: N says nothing about the total increase in the scalefactor between the beginning and end of inflation, which is expected to bemuch larger.40 The assumption here is that inflation stops by the end of slow roll and isfollowed by the field oscillating in an approximately quadratic potential nearthe minimum. The universe eventually reheats to a density greater than thatduring big bang nucleosynthesis, and then the standard thermal history ensues.

While the m2φ2 (p = 2) model was consistent with previousconstraints (Keisler et al. 2011; Sanchez et al. 2012), the currentcombinations of CMB with BAO data place a tight upper limiton r and disfavor the m2φ2 model, which produces predictionsthat fall at the edge of the 95% confidence contour. This modelis allowed by CMB or CMB+H0. Models with smaller valuesof p are consistent with the data, as shown in Figure 10.

The exponential models lead to ns and r predictions thatare independent of scale and therefore independent of N. Thepredictions, as p varies, form a line in the ns–r plane. Thiswhole class of models is allowed at 95% confidence for a rangeof p by the CMB+H0 data, and is excluded (>95% C.L.) by theCMB+BAO and CMB+H0+BAO data.

The potential in hilltop models has the shape of symmetry-breaking potentials that drive φ away from the origin. Thegeneric form of the hilltop potential is V (φ) ∝ 1 − (φ/μ)p.Such models, for a fixed p, have three free parameters: theproportionality constant, μ, and φend. The first two are foundin the potential. The third is needed because the potential doesnot naturally lead to an end to inflation; without making φendexplicit, the end of inflation depends either on the details ofunspecified higher order terms in the potential or on externalphysics. The proportionality constant is set by the amplitude ofscalar fluctuations Δ2

R , while μ, for fixed φend, is constrained byns and r. In turn, we take φend to be constrained above by μ, oforder the vacuum expectation value of the field.

For p � 2, hilltop models can have the behavior of large-fieldmodels for the range of ns allowed by the data. The behavior ofthe p = 2 case in the large-field (φend = μ) limit is shown as apurple region in Figure 10, and is consistent with the data.

Last, we consider small-field inflation models. Examples ofsmall-field potentials include hilltop potentials with p > 2 (forsome values of μ), the Coleman–Weinberg potential (Coleman& Weinberg 1973; with suitably adjusted parameters), andwarped D-brane inflation (Kachru et al. 2003). Because small-field models predict r � 0.01, the SPT data have no constrainingpower on these models through limits on r. All of these modelsare consistent with the data, as long as they are consistent withthe limits on ns.

Constraints on the scale dependence of the spectral index(dns/d ln k) are considered in the companion paper H12.

7. CONCLUSION

We present a measurement of the CMB temperature powerspectrum from 2540 deg2 of sky observed with the SPT. Theseare the first CMB power spectrum results reported for the fullSPT-SZ survey, which encompasses three times the area used inprevious SPT power spectrum analyses (K11; Reichardt et al.2013) The bandpowers cover the third to ninth acoustic peaks(650 < � < 3000) with sample-variance-limited precision at� < 2900. This measurement represents a significant advanceover previous measurements of the damping tail by ACBAR(Reichardt et al. 2009), QUaD (Brown et al. 2009; Friedmanet al. 2009), ACT (Das et al. 2011b), and SPT (Keisler et al.2011).

We find the SPT bandpowers are well fit by a spatially flatΛCDM cosmology with gravitational lensing by large-scalestructure. We use this SPT measurement to extend the dynamicrange probed by the WMAP power spectrum, thus tighteningparameter constraints in the six-parameter ΛCDM model. Withthe exception of the optical depth τ which is constrained bythe large-scale polarization data from WMAP7, adding the full

17


2.0 2.1 2.2 2.3 2.40.0

0.2

0.4

0.6

0.8

1.0

0.10 0.12 1.03 1.04

650< l <30001500< l <3000 650< l <1500 WMAP7-only

0.7 0.8

0.06 0.08 0.10 0.12 0.140.0

0.2

0.4

0.6

0.8

1.0

0.92 0.94 0.96 0.98 1.00 1.9 2.0 2.1 2.2 2.3 2.4 65 70 75 80

Figure 11. Same ΛCDM-model parameters as Figure 5, except showing the effect of including the SPT bandpowers over sub-sets of the �-range. The constraints forWMAP7-only are shown (red dashed lines). Using the SPT+WMAP7 dataset, the constraints are shown from the full �-range of SPT data (black solid lines), from thehigh-� range 1500 < �SPT < 3000 (purple dot-dot-dashed lines), and from the low-� range 650 < �SPT < 1500 (green dot-dashed lines).


survey SPT bandpowers significantly improves measurementsof all ΛCDM parameters. Most notable is that the measurementof the angular sound horizon, θs , tightens by a factor of 2.7because of the number of acoustic peaks detected at high signal-to-noise. Uncertainties on the other four parameters are reducedby a factor of ∼1.4. The combination of SPT and WMAP7 datais used to constrain ns < 1 at 3.9σ .

We examine constraints on three extensions to the ΛCDMmodel. We first allow for a rescaling of the gravitationallensing potential by a parameter AL. Using CMB data, werule out the no-lensing hypothesis (AL � 0) at 8.1σ , the mostsignificant detection to date using the CMB alone, and measurea lensing amplitude, AL = 0.86+0.15(+0.30)

−0.13(−0.25) (68% and 95% C.L.),consistent with the ΛCDM expectations. We expect the lensing-detection significance to triple in a future analysis of the fullSPT-SZ survey using an optimized four-point lensing estimator,similar to the one applied to one fifth of the survey by vanEngelen et al. (2012).

Second, we allow non-zero mean curvature of the observableuniverse. The low-redshift information encoded in the CMB bygravitational lensing helps to improve constraints on the meancurvature of the observable universe for the ΛCDM+Ωk model.Using the CMB alone, we measure Ωk = −0.003+0.014

−0.018, whichis consistent with a flat universe. Models without dark energyare ruled out at 5.4σ .

Last, we review constraints on the amplitude of tensorperturbations. The combination of SPT+WMAP7 is used toconstrain the tensor-to-scalar ratio to be r < 0.18 with 95%confidence.

Adding low-redshift probes of H0 and BAO further tight-ens these constraints. Combining the CMB and H0 datasetsmildly tightens our parameter constraints, and is fully consis-tent with the CMB constraints. Combining the CMB and BAOdatasets leads to significant improvements in our parameterconstraints. Combining all three datasets produces constraints

that lie close to the CMB+BAO constraints; the combinationof CMB+H0+BAO is used to constrain ns < 1 at 5.7σ inthe ΛCDM model, measure Ωk = −0.0059 ± 0.0040 in theΛCDM+Ωk model, and constrain r < 0.11 at 95% confidencein the ΛCDM+r model. This constraint on r approaches thetheoretical limit of how well tensor perturbations can be con-strained from CMB temperature anisotropy, r < 0.1 (95% C.L.).We compare these constraints on ns and r to the predictions ofsingle-field inflation models and exclude several models withgreater than 95% confidence.

Some tension exists between the six datasets included inthe CMB+BAO+H0 combination for a ΛCDM cosmology.However, we assume the uncertainties reported for each ofthe datasets are correct and combine them to produce manyof the results presented here. We refer the reader to H12for a more detailed discussion of the consistency of thedatasets.

In this paper, we have focused on the amplitude and shapeof the primordial power spectrum of scalar and tensor pertur-bations, as well as the effects of gravitational lensing and cur-vature. Further cosmological implications of the bandpowersfrom the full SPT-SZ survey, including constraints on the neu-trino masses, the dark-energy equation of state, the primordialhelium abundance, and the effective number of neutrino speciesare explored in the companion paper H12.

The first CMB temperature power spectrum from the Plancksatellite is expected to be released in 2013. We expect that,given the small beam of the SPT relative to Planck, the datapresented here should remain the most precise measurement ofsmall, angular-scale anisotropy for 2200 � � < 3000. The cos-mological constraints presented here will tighten significantlywith the with the Planck power spectra. Further improvementwill come with the addition of polarization information fromupcoming experiments including Planck, ACT-Pol, Polarbear,and SPT-Pol, the new polarization-sensitive camera on the SPT.

18


We thank Scott Dodelson, John Peacock, David Baumann,and Antonio Riotto for useful conversations. The SPT issupported by the National Science Foundation through grantANT-0638937, with partial support provided by NSF grantPHY-1125897, the Kavli Foundation, and the Gordon and BettyMoore Foundation. The McGill group acknowledges fundingfrom the National Sciences and Engineering Research Councilof Canada, Canada Research Chairs program, and the CanadianInstitute for Advanced Research. Work at Harvard is supportedby grant AST-1009012. R. Keisler acknowledges support fromNASA Hubble Fellowship grant HF-51275.01, B. A. Bensona KICP Fellowship, M. Dobbs an Alfred P. Sloan ResearchFellowship, O. Zahn a BCCP fellowship, M. Millea and L.Knox a NSF grant 0709498. This research used resources of theNational Energy Research Scientific Computing Center, whichis supported by the Office of Science of the U.S. Departmentof Energy under contract No. DE-AC02-05CH11231, and theresources of the University of Chicago Computing Cooperative(UC3), supported in part by the Open Science Grid, NSF grantNSF PHY 1148698. Some of the results in this paper havebeen derived using the HEALPix (Gorski et al. 2005) package.We acknowledge the use of the Legacy Archive for MicrowaveBackground Data Analysis (LAMBDA). Support for LAMBDAis provided by the NASA Office of Space Science.

Facility: South Pole Telescope

APPENDIX

DEPENDENCE OF SPT CONSTRAINTSON MULTIPOLE RANGE

As discussed in Section 6.6 and shown in Figure 5, theΛCDM constraints tighten and shift from the WMAP7 to theSPT+WMAP7 dataset. We explore how different �-ranges ofthe SPT data drive these changes in Figure 11. The shifts inΩch

2 and ΩΛ (or H0) are driven primarily by the bandpowersat higher multipoles, above � = 1500. As discussed in H12,this preference is driven largely by the sensitivity of the SPTdata to gravitational lensing in the high-� acoustic peaks. Incontrast, ns is constrained largely by the lower multipoles; thebandpowers at higher multipoles prefer slightly higher values ofns. The SPT data prefer higher values of θs than do the WMAP7data; each of the sub-ranges pull θs above the WMAP value, butonly the constraining power of the full dataset pulls θs up to theSPT+WMAP7 value.

REFERENCES

Addison, G. E., Dunkley, J., Hajian, A., et al. 2012, ApJ, 752, 120Albrecht, A. 2011, PhRvL, 107, 151102Albrecht, A., & Steinhardt, P. J. 1982, PhRvL, 48, 1220Anderson, L., Aubourg, E., Bailey, S., et al. 2012, MNRAS, 427, 3435Bardeen, J. M., Steinhardt, P. J., & Turner, M. S. 1983, PhRvD, 28, 679Baumann, D., Jackson, M. G., Adshead, P., et al. 2009, in AIP Conf. Proc. 1141,

CMB Polarization Workshop: Theory and Foregrounds, ed. S. Dodelson, D.Baumann, A. Cooray, J. Dunkley, A. Fraisse, M. G. Jackson, A. Kogut, L.Krauss, M. Zaldarriaga, & K. Smith (Melville, NY: AIP), 10

Blake, C., Kazin, E. A., Beutler, F., et al. 2011, MNRAS, 418, 1707Bleem, L. E., van Engelen, A., Holder, G. P., et al. 2012, ApJL, 753, L9Bond, J. R., Efstathiou, G., & Tegmark, M. 1997, MNRAS, 291, L33Brown, M. L., Ade, P., Bock, J., et al. 2009, ApJ, 705, 978Bucher, M., Goldhaber, A. S., & Turok, N. 1995, PhRvD, 52, 3314Calabrese, E., Slosar, A., Melchiorri, A., Smoot, G. F., & Zahn, O. 2008, PhRvD,

77, 123531

Chiang, H. C., Ade, P. A. R., Barkats, D., et al. 2010, ApJ, 711, 1123Christensen, N., Meyer, R., Knox, L., & Luey, B. 2001, CQGra, 18, 2677Coleman, S., & Weinberg, E. 1973, PhRvD, 7, 1888Copeland, E. J., Kolb, E. W., Liddle, A. R., & Lidsey, J. E. 1993, PhRvL,

71, 219Das, S., Marriage, T. A., Ade, P. A. R., et al. 2011a, ApJ, 729, 62Das, S., Sherwin, B. D., Aguirre, P., et al. 2011b, PhRvL, 107, 021301De Zotti, G., Massardi, M., Negrello, M., & Wall, J. 2010, A&ARv, 18, 1Dodelson, S., Kinney, W. H., & Kolb, E. W. 1997, PhRvD, 56, 3207Dodelson, S., & Knox, L. 2000, PhRvL, 84, 3523Dunkley, J., Hlozek, R., Sievers, J., et al. 2011, ApJ, 739, 52Dunner, R., Hasselfield, M., Marriage, T. A., et al. 2013, ApJ, 762, 10Fendt, W. A., & Wandelt, B. D. 2007a, arXiv:0712.0194Fendt, W. A., & Wandelt, B. D. 2007b, ApJ, 654, 2Finkbeiner, D. P., Davis, M., & Schlegel, D. J. 1999, ApJ, 524, 867Fowler, J. W., Acquaviva, V., Ade, P. A. R., et al. 2010, ApJ, 722, 1148Friedman, R. B., Ade, P., Bock, J., et al. 2009, ApJL, 700, L187Gorski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759Guth, A. H. 1981, PhRvD, 23, 347Guth, A. H., & Pi, S. 1982, PhRvL, 49, 1110Hawking, S. W. 1982, PhLB, 115, 295Hirata, C. M., Ho, S., Padmanabhan, N., Seljak, U., & Bahcall, N. A. 2008,

PhRvD, 78, 043520Hivon, E., Gorski, K. M., Netterfield, C. B., et al. 2002, ApJ, 567, 2Hou, Z., Reichardt, C. L., Story, K. T., et al. 2012, arXiv:1212.6267Kachru, S., Kallosh, R., Linde, A., et al. 2003, JCAP, 10, 013Kamionkowski, M., Kosowsky, A., & Stebbins, A. 1997, PhRvD, 55, 7368Keisler, R., Reichardt, C. L., Aird, K. A., et al. 2011, ApJ, 743, 28Kinney, W. H. 2003, NewAR, 47, 967Kinney, W. H., Kolb, E. W., Melchiorri, A., & Riotto, A. 2008, PhRvD,

78, 087302Knox, L. 2006, PhRvD, 73, 023503Knox, L., & Turner, M. S. 1994, PhRvL, 73, 3347Komatsu, E., Dunkley, J., Nolta, M. R., et al. 2009, ApJS, 180, 330Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18Larson, D., Dunkley, J., Hinshaw, G., et al. 2011, ApJS, 192, 16Lewis, A. 2005, PhRvD, 71, 083008Lewis, A., & Bridle, S. 2002, PhRvD, 66, 103511Lewis, A., & Challinor, A. 2006, PhR, 429, 1Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473Liddle, A. R., & Leach, S. M. 2003, PhRvD, 68, 103503Linde, A. D. 1982, PhLB, 108, 389Lueker, M., Reichardt, C. L., Schaffer, K. K., et al. 2010, ApJ, 719, 1045Marriage, T. A., Baptiste Juin, J., Lin, Y.-T., et al. 2011, ApJ, 731, 100Mukhanov, V. F., & Chibisov, G. V. 1981, JETPL, 33, 532Padmanabhan, N., Xu, X., Eisenstein, D. J., et al. 2012, MNRAS, 427,

2132Reichardt, C. L., Ade, P. A. R., Bock, J. J., et al. 2009, ApJ, 694, 1200Reichardt, C. L., Shaw, L., Zahn, O., et al. 2012, ApJ, 755, 70Reichardt, C. L., Stalder, B., Bleem, L. E., et al. 2013, ApJ, 763, 127Riess, A. G., Macri, L., Casertano, S., et al. 2011, ApJ, 730, 119Rozo, E., Wechsler, R. H., Rykoff, E. S., et al. 2010, ApJ, 708, 645Sanchez, A. G., Scoccola, C. G., Ross, A. J., et al. 2012, MNRAS, 425, 415Schaffer, K. K., Crawford, T. M., Aird, K. A., et al. 2011, ApJ, 743, 90Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525Seljak, U., & Zaldarriaga, M. 1997, PhRvL, 78, 2054Shaw, L. D., Nagai, D., Bhattacharya, S., & Lau, E. T. 2010, ApJ, 725,

1452Sherwin, B. D., Das, S., Hajian, A., et al. 2012, PhRvD, 86, 083006Sherwin, B. D., Dunkley, J., Das, S., et al. 2011, PhRvL, 107, 021302Shirokoff, E., Reichardt, C. L., Shaw, L., et al. 2011, ApJ, 736, 61Silk, J. 1968, ApJ, 151, 459Smith, K. M., Zahn, O., & Dore, O. 2007, PhRvD, 76, 043510Snyder, J. P. 1987, Map Projections–A Working Manual (Washington: U.S.

Geological Survey)Starobinsky, A. A. 1982, PhLB, 117, 175van Engelen, A., Keisler, R., Zahn, O., et al. 2012, ApJ, 756, 142Vieira, J. D., Crawford, T. M., Switzer, E. R., et al. 2010, ApJ, 719, 763Vikhlinin, A., Kravtsov, A. V., Burenin, R. A., et al. 2009, ApJ, 692, 1060White, S., Navarro, J., Evrard, A., & Frenk, C. 1993, Natur, 366, 429Williamson, R., Benson, B. A., High, F. W., et al. 2011, ApJ, 738, 139Zahn, O., Reichardt, C. L., Shaw, L., et al. 2012, ApJ, 756, 65Zaldarriaga, M., Spergel, D. N., & Seljak, U. 1997, ApJ, 488, 1

19

http://dx.doi.org/10.1088/0004-637X/752/2/120

http://adsabs.harvard.edu/abs/2012ApJ...752..120A

http://adsabs.harvard.edu/abs/2012ApJ...752..120A

http://adsabs.harvard.edu/abs/2011PhRvL.107o1102A

http://adsabs.harvard.edu/abs/2011PhRvL.107o1102A

http://adsabs.harvard.edu/abs/1982PhRvL..48.1220A

http://adsabs.harvard.edu/abs/1982PhRvL..48.1220A

http://dx.doi.org/10.1111/j.1365-2966.2012.22066.x

http://adsabs.harvard.edu/abs/2012MNRAS.427.3435A

http://adsabs.harvard.edu/abs/2012MNRAS.427.3435A

http://adsabs.harvard.edu/abs/1983PhRvD..28..679B

http://adsabs.harvard.edu/abs/1983PhRvD..28..679B

http://adsabs.harvard.edu/abs/2009AIPC.1141...10B

http://dx.doi.org/10.1111/j.1365-2966.2011.19592.x

http://adsabs.harvard.edu/abs/2011MNRAS.418.1707B

http://adsabs.harvard.edu/abs/2011MNRAS.418.1707B

http://dx.doi.org/10.1088/2041-8205/753/1/L9

http://adsabs.harvard.edu/abs/2012ApJ...753L...9B

http://adsabs.harvard.edu/abs/2012ApJ...753L...9B

http://adsabs.harvard.edu/abs/1997MNRAS.291L..33B

http://adsabs.harvard.edu/abs/1997MNRAS.291L..33B

http://dx.doi.org/10.1088/0004-637X/705/1/978

http://adsabs.harvard.edu/abs/2009ApJ...705..978B

http://adsabs.harvard.edu/abs/2009ApJ...705..978B

http://adsabs.harvard.edu/abs/1995PhRvD..52.3314B

http://adsabs.harvard.edu/abs/1995PhRvD..52.3314B

http://adsabs.harvard.edu/abs/2008PhRvD..77l3531C

http://adsabs.harvard.edu/abs/2008PhRvD..77l3531C

http://dx.doi.org/10.1088/0004-637X/711/2/1123

http://adsabs.harvard.edu/abs/2010ApJ...711.1123C

http://adsabs.harvard.edu/abs/2010ApJ...711.1123C

http://dx.doi.org/10.1088/0264-9381/18/14/306

http://adsabs.harvard.edu/abs/2001CQGra..18.2677C

http://adsabs.harvard.edu/abs/2001CQGra..18.2677C

http://adsabs.harvard.edu/abs/1973PhRvD...7.1888C

http://adsabs.harvard.edu/abs/1973PhRvD...7.1888C

http://adsabs.harvard.edu/abs/1993PhRvL..71..219C

http://adsabs.harvard.edu/abs/1993PhRvL..71..219C

http://dx.doi.org/10.1088/0004-637X/729/1/62

http://adsabs.harvard.edu/abs/2011ApJ...729...62D


http://adsabs.harvard.edu/abs/2011PhRvL.107b1301D

http://adsabs.harvard.edu/abs/2011PhRvL.107b1301D

http://adsabs.harvard.edu/abs/2010A&ARv..18....1D

http://adsabs.harvard.edu/abs/2010A&ARv..18....1D

http://adsabs.harvard.edu/abs/1997PhRvD..56.3207D

http://adsabs.harvard.edu/abs/1997PhRvD..56.3207D

http://adsabs.harvard.edu/abs/2000PhRvL..84.3523D

http://adsabs.harvard.edu/abs/2000PhRvL..84.3523D

http://dx.doi.org/10.1088/0004-637X/739/1/52



http://dx.doi.org/10.1088/0004-637X/762/1/10



http://www.arxiv.org/abs/0712.0194

http://dx.doi.org/10.1086/508342

http://adsabs.harvard.edu/abs/2007ApJ...654....2F

http://adsabs.harvard.edu/abs/2007ApJ...654....2F

http://dx.doi.org/10.1086/307852

http://adsabs.harvard.edu/abs/1999ApJ...524..867F

http://adsabs.harvard.edu/abs/1999ApJ...524..867F

http://dx.doi.org/10.1088/0004-637X/722/2/1148

http://adsabs.harvard.edu/abs/2010ApJ...722.1148F

http://adsabs.harvard.edu/abs/2010ApJ...722.1148F

http://dx.doi.org/10.1088/0004-637X/700/2/L187

http://adsabs.harvard.edu/abs/2009ApJ...700L.187F

http://adsabs.harvard.edu/abs/2009ApJ...700L.187F

http://dx.doi.org/10.1086/427976

http://adsabs.harvard.edu/abs/2005ApJ...622..759G

http://adsabs.harvard.edu/abs/2005ApJ...622..759G

http://adsabs.harvard.edu/abs/1981PhRvD..23..347G

http://adsabs.harvard.edu/abs/1981PhRvD..23..347G

http://adsabs.harvard.edu/abs/1982PhRvL..49.1110G

http://adsabs.harvard.edu/abs/1982PhRvL..49.1110G

http://adsabs.harvard.edu/abs/1982PhLB..115..295H

http://adsabs.harvard.edu/abs/1982PhLB..115..295H

http://adsabs.harvard.edu/abs/2008PhRvD..78d3520H

http://adsabs.harvard.edu/abs/2008PhRvD..78d3520H

http://dx.doi.org/10.1086/338126

http://adsabs.harvard.edu/abs/2002ApJ...567....2H

http://adsabs.harvard.edu/abs/2002ApJ...567....2H

http://www.arxiv.org/abs/1212.6267

http://dx.doi.org/10.1088/1475-7516/2003/10/013

http://adsabs.harvard.edu/abs/2003JCAP...10..013K

http://adsabs.harvard.edu/abs/2003JCAP...10..013K

http://adsabs.harvard.edu/abs/1997PhRvD..55.7368K

http://adsabs.harvard.edu/abs/1997PhRvD..55.7368K

http://dx.doi.org/10.1088/0004-637X/743/1/28

http://adsabs.harvard.edu/abs/2011ApJ...743...28K

http://adsabs.harvard.edu/abs/2011ApJ...743...28K

http://dx.doi.org/10.1016/j.newar.2003.09.028

http://adsabs.harvard.edu/abs/2003NewAR..47..967K

http://adsabs.harvard.edu/abs/2003NewAR..47..967K

http://adsabs.harvard.edu/abs/2008PhRvD..78h7302K

http://adsabs.harvard.edu/abs/2008PhRvD..78h7302K

http://adsabs.harvard.edu/abs/2006PhRvD..73b3503K

http://adsabs.harvard.edu/abs/2006PhRvD..73b3503K

http://adsabs.harvard.edu/abs/1994PhRvL..73.3347K

http://adsabs.harvard.edu/abs/1994PhRvL..73.3347K

http://dx.doi.org/10.1088/0067-0049/180/2/330

http://adsabs.harvard.edu/abs/2009ApJS..180..330K

http://adsabs.harvard.edu/abs/2009ApJS..180..330K

http://dx.doi.org/10.1088/0067-0049/192/2/18

http://adsabs.harvard.edu/abs/2011ApJS..192...18K

http://adsabs.harvard.edu/abs/2011ApJS..192...18K

http://dx.doi.org/10.1088/0067-0049/192/2/16

http://adsabs.harvard.edu/abs/2011ApJS..192...16L

http://adsabs.harvard.edu/abs/2011ApJS..192...16L

http://adsabs.harvard.edu/abs/2005PhRvD..71h3008L

http://adsabs.harvard.edu/abs/2005PhRvD..71h3008L

http://adsabs.harvard.edu/abs/2002PhRvD..66j3511L


http://adsabs.harvard.edu/abs/2006PhR...429....1L

http://adsabs.harvard.edu/abs/2006PhR...429....1L

http://dx.doi.org/10.1086/309179

http://adsabs.harvard.edu/abs/2000ApJ...538..473L

http://adsabs.harvard.edu/abs/2000ApJ...538..473L



http://adsabs.harvard.edu/abs/1982PhLB..108..389L

http://adsabs.harvard.edu/abs/1982PhLB..108..389L

http://dx.doi.org/10.1088/0004-637X/719/2/1045

http://adsabs.harvard.edu/abs/2010ApJ...719.1045L

http://adsabs.harvard.edu/abs/2010ApJ...719.1045L

http://dx.doi.org/10.1088/0004-637X/731/2/100

http://adsabs.harvard.edu/abs/2011ApJ...731..100M

http://adsabs.harvard.edu/abs/2011ApJ...731..100M

http://adsabs.harvard.edu/abs/1981JETPL..33..532M

http://adsabs.harvard.edu/abs/1981JETPL..33..532M

http://dx.doi.org/10.1111/j.1365-2966.2012.21888.x

http://adsabs.harvard.edu/abs/2012MNRAS.427.2132P

http://adsabs.harvard.edu/abs/2012MNRAS.427.2132P

http://dx.doi.org/10.1088/0004-637X/694/2/1200

http://adsabs.harvard.edu/abs/2009ApJ...694.1200R

http://adsabs.harvard.edu/abs/2009ApJ...694.1200R

http://dx.doi.org/10.1088/0004-637X/755/1/70

http://adsabs.harvard.edu/abs/2012ApJ...755...70R

http://adsabs.harvard.edu/abs/2012ApJ...755...70R

http://dx.doi.org/10.1088/0004-637X/763/2/127

http://adsabs.harvard.edu/abs/2013ApJ...763..127R


http://dx.doi.org/10.1088/0004-637X/730/2/119



http://dx.doi.org/10.1088/0004-637X/708/1/645



http://dx.doi.org/10.1111/j.1365-2966.2012.21502.x

http://adsabs.harvard.edu/abs/2012MNRAS.425..415S

http://adsabs.harvard.edu/abs/2012MNRAS.425..415S

http://dx.doi.org/10.1088/0004-637X/743/1/90

http://adsabs.harvard.edu/abs/2011ApJ...743...90S


http://dx.doi.org/10.1086/305772

http://adsabs.harvard.edu/abs/1998ApJ...500..525S


http://adsabs.harvard.edu/abs/1997PhRvL..78.2054S

http://adsabs.harvard.edu/abs/1997PhRvL..78.2054S

http://dx.doi.org/10.1088/0004-637X/725/2/1452

http://adsabs.harvard.edu/abs/2010ApJ...725.1452S

http://adsabs.harvard.edu/abs/2010ApJ...725.1452S

http://adsabs.harvard.edu/abs/2012PhRvD..86h3006S

http://adsabs.harvard.edu/abs/2012PhRvD..86h3006S

http://adsabs.harvard.edu/abs/2011PhRvL.107b1302S

http://adsabs.harvard.edu/abs/2011PhRvL.107b1302S

http://dx.doi.org/10.1088/0004-637X/736/1/61



http://dx.doi.org/10.1086/149449



http://adsabs.harvard.edu/abs/2007PhRvD..76d3510S

http://adsabs.harvard.edu/abs/2007PhRvD..76d3510S

http://adsabs.harvard.edu/abs/1982PhLB..117..175S

http://adsabs.harvard.edu/abs/1982PhLB..117..175S

http://dx.doi.org/10.1088/0004-637X/756/2/142

http://adsabs.harvard.edu/abs/2012ApJ...756..142V


http://dx.doi.org/10.1088/0004-637X/719/1/763



http://dx.doi.org/10.1088/0004-637X/692/2/1060

http://adsabs.harvard.edu/abs/2009ApJ...692.1060V

http://adsabs.harvard.edu/abs/2009ApJ...692.1060V

http://dx.doi.org/10.1038/366429a0

http://adsabs.harvard.edu/abs/1993Natur.366..429W

http://adsabs.harvard.edu/abs/1993Natur.366..429W

http://dx.doi.org/10.1088/0004-637X/738/2/139

http://adsabs.harvard.edu/abs/2011ApJ...738..139W

http://adsabs.harvard.edu/abs/2011ApJ...738..139W

http://dx.doi.org/10.1088/0004-637X/756/1/65

http://adsabs.harvard.edu/abs/2012ApJ...756...65Z

http://adsabs.harvard.edu/abs/2012ApJ...756...65Z

http://dx.doi.org/10.1086/304692

http://adsabs.harvard.edu/abs/1997ApJ...488....1Z

http://adsabs.harvard.edu/abs/1997ApJ...488....1Z

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